Global well-posedness and scattering for the energy-critical

Annals of Mathematics, 167 (2008), 767–865

Global well-posedness and scattering

for the energy-critical nonlinearSchrodinger equation in R3

By J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao*

Abstract

We obtain global well-posedness, scattering, and global L10t,x spacetime

bounds for energy-class solutions to the quintic defocusing Schrodinger equa-tion in R1+3, which is energy-critical. In particular, this establishes globalexistence of classical solutions. Our work extends the results of Bourgain [4]and Grillakis [20], which handled the radial case. The method is similar inspirit to the induction-on-energy strategy of Bourgain [4], but we perform theinduction analysis in both frequency space and physical space simultaneously,and replace the Morawetz inequality by an interaction variant (first used in[12], [13]). The principal advantage of the interaction Morawetz estimate isthat it is not localized to the spatial origin and so is better able to handlenonradial solutions. In particular, this interaction estimate, together with analmost-conservation argument controlling the movement of L2 mass in fre-quency space, rules out the possibility of energy concentration.

Contents

1. Introduction1.1. Critical NLS and main result1.2. Notation

2. Local conservation laws

3. Review of Strichartz theory in R1+3

3.1. Linear Strichartz estimates3.2. Bilinear Strichartz estimates3.3. Quintilinear Strichartz estimates3.4. Local well-posedness and perturbation theory

*J.C. is supported in part by N.S.F. Grant DMS-0100595, N.S.E.R.C. Grant R.G.P.I.N.250233-03 and the Sloan Foundation. M.K. was supported in part by N.S.F. Grant DMS-0303704; and by the McKnight and Sloan Foundations. G.S. is supported in part by N.S.F.Grant DMS-0100375, N.S.F. Grant DMS-0111298 through the IAS, and the Sloan Founda-tion. H.T. is supported in part by J.S.P.S. Grant No. 15740090 and by a J.S.P.S. PostdoctoralFellowship for Research Abroad. T.T. is a Clay Prize Fellow and is supported in part bygrants from the Packard Foundation.

768 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO

4. Overview of proof of global spacetime bounds4.1. Zeroth stage: Induction on energy4.2. First stage: Localization control on u4.3. Second stage: Localized Morawetz estimate4.4. Third stage: Nonconcentration of energy

5. Frequency delocalized at one time =⇒ spacetime bounded6. Small L6

x norm at one time =⇒ spacetime bounded7. Spatial concentration of energy at every time8. Spatial delocalized at one time =⇒ spacetime bounded9. Reverse Sobolev inequality

10. Interaction Morawetz: generalities10.1. Virial-type identity10.2. Interaction virial identity and general interaction Morawetz estimate

for general equations11. Interaction Morawetz: The setup and an averaging argument12. Interaction Morawetz: Strichartz control13. Interaction Morawetz: Error estimate14. Interaction Morawetz: A double Duhamel trick15. Preventing energy evacuation

15.1. The setup and contradiction argument15.2. Spacetime estimates for high, medium, and low frequencies15.3. Controlling the localized L2 mass increment

16. Remarks

References

1. Introduction

1.1. Critical NLS and main result. We consider the Cauchy problem forthe quintic defocusing Schrodinger equation in R1+3{

iut + Δu = |u|4uu(0, x) = u0(x),

(1.1)

where u(t, x) is a complex-valued field in spacetime Rt × R3x. This equation

has as Hamiltonian,

E(u(t)) :=∫

12|∇u(t, x)|2 +

16|u(t, x)|6 dx.(1.2)

Since the Hamiltonian (1.2) is preserved by the flow (1.1) we shall often referto it as the energy and write E(u) for E(u(t)).

Semilinear Schrodinger equations - with and without potentials, and withvarious nonlinearities - arise as models for diverse physical phenomena, includ-ing Bose-Einstein condensates [23], [35] and as a description of the envelopedynamics of a general dispersive wave in a weakly nonlinear medium (see e.g.

SCATTERING FOR 3D CRITICAL NLS 769

the survey in [43], Chapter 1). Our interest here in the defocusing quinticequation (1.1) is motivated mainly, though, by the fact that the problem iscritical with respect to the energy norm. Specifically, we map a solution toanother solution through the scaling u �→ uλ defined by

uλ(t, x) :=1

λ1/2u

(t

λ2,x

λ

),(1.3)

and this scaling leaves both terms in the energy invariant.The Cauchy problem for this equation has been intensively studied ([9],

[20], [4], [5],[18], [26]). It is known (see e.g. [10], [9]) that if the initial data u0(x)has finite energy, then the Cauchy problem is locally well-posed, in the sensethat there exists a local-in-time solution to (1.1) which lies in C0

t H1x ∩ L10

t,x,and is unique in this class; furthermore the map from initial data to solu-tion is locally Lipschitz continuous in these norms. If the energy is small,then the solution is known to exist globally in time, and scatters to a solutionu±(t) to the free Schrodinger equation (i∂t + Δ)u± = 0, in the sense that‖u(t) − u±(t)‖H1(R3) → 0 as t → ±∞. For (1.1) with large initial data, thearguments in [10], [9] do not extend to yield global well-posedness, even withthe conservation of the energy (1.2), because the time of existence given by thelocal theory depends on the profile of the data as well as on the energy.1 Forlarge finite energy data which is assumed to be in addition radially symmet-ric, Bourgain [4] proved global existence and scattering for (1.1) in H1(R3).Subsequently Grillakis [20] gave a different argument which recovered part of[4] — namely, global existence from smooth, radial, finite energy data. Forgeneral large data — in particular, general smooth data — global existenceand scattering were open.

Our main result is the following global well-posedness result for (1.1) inthe energy class.

Theorem 1.1. For any u0 with finite energy, E(u0) < ∞, there exists aunique2 global solution u ∈ C0

t (H1x) ∩ L10

t,x to (1.1) such that∫ ∞

−∞

∫R3

|u(t, x)|10 dxdt ≤ C(E(u0)).(1.4)

for some constant C(E(u0)) that depends only on the energy.

1This is in constrast with sub-critical equations such as the cubic equation iut + Δu =|u|2u, for which one can use the local well-posedness theory to yield global well-posednessand scattering even for large energy data (see [17], and the surveys [7], [8]).

2In fact, uniqueness actually holds in the larger space C0t (H1

x) (thus eliminating the con-straint that u ∈ L10

t,x), as one can show by adapting the arguments of [27], [15], [14]; seeSection 16.


As is well-known (see e.g. [5], or [13] for the sub-critical analogue), theL10

t,x bound above also gives scattering, asymptotic completeness, and uniformregularity:

Corollary 1.2. Let u0 have finite energy. Then there exist finite energysolutions u±(t, x) to the free Schrodinger equation (i∂t + Δ)u± = 0 such that

‖u±(t) − u(t)‖H1 → 0 as t → ±∞.

Furthermore, the maps u0 �→ u±(0) are homeomorphisms from H1(R3) toH1(R3). Finally, if u0 ∈ Hs for some s > 1, then u(t) ∈ Hs for all time t,and one has the uniform bounds

supt∈R

‖u(t)‖Hs ≤ C(E(u0), s)‖u0‖Hs .

It is also fairly standard to show that the L10t,x bound (1.4) implies further

spacetime integrability on u. For instance u obeys all the Strichartz estimatesthat a free solution with the same regularity does (see, for example, Lemma3.12 below).

The results here have analogs in previous work on second order wave equa-tions on R3+1 with energy-critical (quintic) defocusing nonlinearities. Global-in-time existence for such equations from smooth data was shown by Grillakis[21], [22] (for radial data see Struwe [42], for small energy data see Rauch [36]);global-in-time solutions from finite energy data were shown in Kapitanski [25],Shatah-Struwe [39]. For an analog of the scattering statement in Corollary 1.2for the critical wave equation; see Bahouri-Shatah [2], Bahouri-Gerard [1] forthe scattering statement for Klein-Gordon equations see Nakanishi [30] (forradial data, see Ginibre-Soffer-Velo[16]). The existence results mentioned hereall involve an argument showing that the solution’s energy cannot concentrate.These energy nonconcentration proofs combine Morawetz inequalities (a prioriestimates for the nonlinear equations which bound some quantity that scaleslike energy) with careful analysis that strengthens the Morawetz bound tocontrol of energy. Besides the presence of infinite propagation speeds, a maindifference between (1.1) and the hyperbolic analogs is that here time scaleslike λ2, and as a consequence the quantity bounded by the Morawetz estimateis supercritical with respect to energy.

Section 4 below provides a fairly complete outline of the proof of Theo-rem 1.1. In this introduction we only briefly sketch some of the ideas involved:a suitable modification of the Morawetz inequality for (1.1), along with thefrequency-localized L2 almost-conservation law that we’ll ultimately use toprohibit energy concentration.


A typical example of a Morawetz inequality for (1.1) is the following bounddue to Lin and Strauss [33] who cite [34] as motivation,∫

I

∫R3

|u(t, x)|6|x| dxdt �

(supt∈I

‖u(t)‖H1/2

)2

(1.5)

for arbitrary time intervals I. (The estimate (1.5) follows from a computationshowing the quantity, ∫

R3

Im(

u∇u · x

|x|

)dx(1.6)

is monotone in time.) Observe that the right-hand side of (1.5) will not growin I if the H1 and L2 norms are bounded, and so this estimate gives a uni-form bound on the left-hand side where I is any interval on which we knowthe solution exists. However, in the energy-critical problem (1.1) there aretwo drawbacks with this estimate. The first is that the right-hand side in-volves the H1/2 norm, instead of the energy E. This is troublesome sinceany Sobolev norm rougher than H1 is supercritical with respect to the scaling(1.3). Specifically, the right-hand side of (1.5) increases without bound whenwe simply scale given finite energy initial data according to (1.3) with λ large.The second difficulty is that the left-hand side is localized near the spatial ori-gin x = 0 and does not convey as much information about the solution u awayfrom this origin. To get around the first difficulty Bourgain [4] and Grillakis[20] introduced a localized variant of the above estimate:∫

I

∫|x|�|I|1/2

|u(t, x)|6|x| dxdt � E(u)|I|1/2.(1.7)

As an example of the usefulness of (1.7), we observe that this estimate prohibitsthe existence of finite energy (stationary) pseudosoliton solutions to (1.1). Bya (stationary) pseudosoliton we mean a solution such that |u(t, x)| ∼ 1 for allt ∈ R and |x| � 1; this notion includes soliton and breather type solutions.Indeed, applying (1.7) to such a solution, we would see that the left-hand sidegrows by at least |I|, while the right-hand side is O(|I| 12 ), and so a pseudosoli-ton solution will lead to a contradiction for |I| sufficiently large. A similarargument allows one to use (1.7) to prevent “sufficiently rapid” concentrationof (potential) energy at the origin; for instance, (1.7) can also be used to ruleout self-similar type blowup,3, where the potential energy density |u|6 concen-trates in the ball |x| < A|t − t0| as t → t−0 for some fixed A > 0. In [4],one main use of (1.7) was to show that for each fixed time interval I, there

3This is not the only type of self-similar blowup scenario; another type is when the energyconcentrates in a ball |x| ≤ A|t − t0|1/2 as t → t−0 . This type of blowup is consistent withthe scaling (1.3) and is not directly ruled out by (1.7); however it can instead be ruled outby spatially local mass conservation estimates. See [4], [20]


exists at least one time t0 ∈ I for which the potential energy was dispersed atscale |I|1/2 or greater (i.e. the potential energy could not concentrate on a ball|x| � |I|1/2 for all times in I).

To summarize, the localized Morawetz estimate (1.7) is very good at pre-venting u from concentrating near the origin; this is especially useful in thecase of radial solutions u, since the radial symmetry (combined with conser-vation of energy) enforces decay of u away from the origin, and so resolvesthe second difficulty with the Morawetz estimate mentioned earlier. However,the estimate is less useful when the solution is allowed to concentrate awayfrom the origin. For instance, if we aim to preclude the existence of a movingpseudosoliton solution, in which |u(t, x)| ∼ 1 when |x − vt| � 1 for some fixedvelocity v, then the left-hand side of (1.7) only grows like log |I| and so onedoes not necessarily obtain a contradiction.4

It is thus of interest to remove the 1/|x| denominator in (1.5), (1.7), so thatthese estimates can more easily prevent concentration at arbitrary locationsin spacetime. In [12], [13] this was achieved by translating the origin in theintegrand of (1.6) to an arbitrary point y, and averaging against the L1 massdensity |u(y)|2 dy. In particular, the following interaction Morawetz estimate5∫

I

∫R3

|u(t, x)|4 dxdt � ‖u(0)‖2L2

(supt∈I

‖u(t)‖H1/2

)2

(1.8)

was obtained. (We have since learned that this averaging argument has ananalog in early work presenting and analyzing interaction functionals for onedimensional hyperbolic systems, e.g. [19], [38].) This L4

t,x estimate alreadygives a short proof of scattering in the energy class (and below!) for thecubic nonlinear Schrodinger equation (see [12], [13]); however, like (1.5), thisestimate is not suitable for the critical problem because the right-hand side isnot controlled by the energy E(u). One could attempt to localize (1.8) as in(1.7), obtaining for instance a scale-invariant estimate such as∫

I

∫|x|�|I|1/2

|u(t, x)|4 dxdt � E(u)2|I|3/2,(1.9)

4At first glance it may appear that the global estimate (1.5) is still able to preclude theexistence of such a pseudosoliton, since the right-hand side does not seem to grow much as Igets larger. This can be done in the cubic problem (see e.g. [17]) but in the critical problemone can lose control of the H1/2 norm, by adding some very low frequency components tothe soliton solution u. One might object that one could use L2 conservation to control theH1/2 norm, however one can rescale the solution to make the L2 norm (and hence the H1/2

norm) arbitrarily large.5Strictly speaking, in [12], [13] this estimate was obtained for the cubic defocusing non-

linear Schrodinger equation instead of the quintic, but the argument in fact works for allnonlinear Schrodinger equations with a pure power defocusing nonlinearity, and even fora slightly more general class of repulsive nonlinearities satisfying a standard monotonicitycondition. See [13] and Section 10 below for more discussion.


but this estimate, while true (in fact it follows immediately from Sobolev andHolder), is useless for such purposes as prohibiting soliton-like behaviour, sincethe left-hand side grows like |I| while the right-hand side grows like |I|3/2. Noris this estimate useful for preventing any sort of energy concentration.

Our solution to these difficulties proceeds in the context of an induction-on-energy argument as in [4]: assume for contradiction that Theorem 1.1 isfalse, and consider a solution of minimal energy among all those solutions withL10

x,t norm above some threshhold. We first show, without relying on any ofthe above Morawetz-type inequalities, that such a minimal energy blowup so-lution would have to be localized in both frequency and in space at all times.Second, we prove that this localized blowup solution satisfies Proposition 4.9,which localizes (1.8) in frequency rather than in space. Roughly speaking,the frequency localized Morawetz inequality of Proposition 4.9 states that af-ter throwing away some small energy, low frequency portions of the blow-upsolution, the remainder obeys good L4

t,x estimates. In principle, this estimateshould follow simply by repeating the proof of (1.8) with u replaced by the highfrequency portion of the solution, and then controlling error terms; howeversome of the error terms are rather difficult and the proof of the frequency-localized Morawetz inequality is quite technical. We emphasize that, unlikethe estimates (1.5), (1.7), (1.8), the frequency-localized Morawetz inequality(4.19) is not an a priori estimate valid for all solutions of (1.1), but insteadapplies only to minimal energy blowup solutions; see Section 4 for furtherdiscussion and precise definitions.

The strategy is then to try to use Sobolev embedding to boost this L4t,x

control to L10t,x control which would contradict the existence of the blow-up so-

lution. There is, however, a remaining enemy, which is that the solution mayshift its energy from low frequencies to high, possibly causing the L10

t,x norm toblow up while the L4

t,x norm stays bounded. To prevent this we look at whatsuch a frequency evacuation would imply for the location — in frequency space— of the blow-up solution’s L2 mass. Specifically, we prove a frequency local-ized L2 mass estimate that gives us information for longer time intervals thanseem to be available from the spatially localized mass conservation laws usedin the previous radial work ([4], [20]). By combining this frequency localizedmass estimate with the L4

t,x bound and plenty of Strichartz estimate analysis,we can control the movement of energy and mass from one frequency rangeto another, and prevent the low-to-high cascade from occurring. The argu-ment here is motivated by our previous low-regularity work involving almostconservation laws (e.g. [13]).

The remainder of the paper is organized as follows: Section 2 reviewssome simple, classical conservation laws for Schrodinger equations which willbe used througout, but especially in proving the frequency localized interac-tion Morawetz estimate. In Section 3 we recall some linear and multilinear


Strichartz estimates, along with the useful nonlinear perturbation statementof Lemma 3.10. Section 4 outlines in some detail the argument behind ourmain Theorem, leaving the proofs of each step to Sections 5–15 of the pa-per. Section 16 presents some miscellaneous remarks, including a proof of theunconditional uniqueness statement alluded to above.

Acknowledgements. We thank the Institute for Mathematics and itsApplications (IMA) for hosting our collaborative meeting in July 2002. Wethank Andrew Hassell, Sergiu Klainerman, and Jalal Shatah for interestingdiscussions related to the interaction Morawetz estimate, and Jean Bourgainfor valuable comments on an early draft of this paper, to Monica Visan and theanonymous referee for their thorough reading of the manuscript and for manyimportant corrections, and to Changxing Miao and Guixiang Xu for furthercorrections. We thank Manoussos Grillakis for explanatory details related to[20]. Finally, it will be clear to the reader that our work here relies heavily inplaces on arguments developed by J. Bourgain in [4].

1.2. Notation. If X, Y are nonnegative quantities, we use X � Y orX = O(Y ) to denote the estimate X ≤ CY for some C (which may depend onthe critical energy Ecrit (see Section 4) but not on any other parameter suchas η), and X ∼ Y to denote the estimate X � Y � X. We use X � Y tomean X ≤ cY for some small constant c (which is again allowed to depend onEcrit).

We use C 1 to denote various large finite constants, and 0 < c � 1 todenote various small constants.

The Fourier transform on R3 is defined by

f(ξ) :=∫

R3

e−2πix·ξf(x) dx,

giving rise to the fractional differentiation operators |∇|s, 〈∇〉s defined by

|∇|sf(ξ) := |ξ|sf(ξ); 〈∇〉sf(ξ) := 〈ξ〉sf(ξ)

where 〈ξ〉 := (1 + |ξ|2)1/2. In particular, we will use ∇ to denote the spatialgradient ∇x. This in turn defines the Sobolev norms

‖f‖Hs(R3) := ‖|∇|sf‖L2(R3); ‖f‖Hs(R3) := ‖〈∇〉sf‖L2(R3).

More generally we define

‖f‖W s,p(R3) := ‖|∇|sf‖Lp(R3); ‖f‖W s,p(R3) := ‖〈∇〉sf‖Lp(R3)

for s ∈ R and 1 < p < ∞.We let eitΔ be the free Schrodinger propagator; in terms of the Fourier

transform, this is given by

eitΔf(ξ) = e−4π2it|ξ|2 f(ξ)(1.10)


while in physical space we have

eitΔf(x) =1

(4πit)3/2

∫R3

ei|x−y|2/4tf(y) dy(1.11)

for t �= 0, using an appropriate branch cut to define the complex square root. Inparticular the propagator preserves all the Sobolev norms Hs(R3) and Hs(R3),and also obeys the dispersive inequality

‖eitΔf‖L∞x (R3) � |t|−3/2‖f‖L1

x(R3).(1.12)

We also record Duhamel ’s formula

u(t) = ei(t−t0)Δu(t0) − i

∫ t

t0

ei(t−s)Δ(iut + Δu)(s) ds(1.13)

for any Schwartz u and any times t0, t ∈ R, with the convention that∫ tt0

= −∫ t0t

if t < t0.We use the notation O(X) to denote an expression which is schemati-

cally of the form X; this means that O(X) is a finite linear combination ofexpressions which look like X but with some factors possibly replaced by theircomplex conjugates. Thus for instance 3u2v2|v|2+9|u|2|v|4+3u2v2|v|2 qualifiesto be of the form O(u2v4), and similarly we have

|u + v|6 = |u|6 + |v|6 +5∑

j=1

O(ujv6−j)(1.14)

and

|u + v|4(u + v) = |u|4u + |v|4v +4∑

j=1

O(ujv5−j).(1.15)

We will sometimes denote partial derivatives using subscripts: ∂xju =

∂ju = uj . We will also implicitly use the summation convention when indicesare repeated in expressions below.

We shall need the following Littlewood-Paley projection operators. Letϕ(ξ) be a bump function adapted to the ball {ξ ∈ R3 : |ξ| ≤ 2} which equals1 on the ball {ξ ∈ R3 : |ξ| ≤ 1}. Define a dyadic number to be any numberN ∈ 2Z of the form N = 2j where j ∈ Z is an integer. For each dyadic numberN , we define the Fourier multipliers

P≤Nf(ξ) := ϕ(ξ/N)f(ξ)

P>Nf(ξ) := (1 − ϕ(ξ/N))f(ξ)

PNf(ξ) := (ϕ(ξ/N) − ϕ(2ξ/N))f(ξ).

We similarly define P<N and P≥N . Note in particular the telescoping identities

P≤Nf =∑

M≤N

PMf ; P>Nf =∑

M>N

PMf ; f =∑M

PMf


for all Schwartz f , where M ranges over dyadic numbers. We also define

PM<·≤N := P≤N − P≤M =∑

M<N ′≤N

PN ′

whenever M ≤ N are dyadic numbers. Similarly define PM≤·≤N , etc.The symbol u shall always refer to a solution to the nonlinear Schrodinger

equation (1.1). We shall use uN to denote the frequency piece uN := PNu

of u, and similarly define u≥N = P≥Nu, etc. While this may cause someconfusion with the notation uj used to denote derivatives of u, the meaning ofthe subscript should be clear from context.

The Littlewood-Paley operators commute with derivative operators (in-cluding |∇|s and i∂t+Δ), the propagator eitΔ, and conjugation operations, areself-adjoint, and are bounded on every Lebesgue space Lp and Sobolev spaceHs (if 1 ≤ p ≤ ∞, of course). Furthermore, they obey the following easily ver-ified Sobolev (and Bernstein) estimates for R3 with s ≥ 0 and 1 ≤ p ≤ q ≤ ∞:

‖P≥Nf‖Lp � N−s‖|∇|sP≥Nf‖Lp ,(1.16)

‖P≤N |∇|sf‖Lp � N s‖P≤Nf‖Lp ,(1.17)

‖PN |∇|±sf‖Lp ∼ N±s‖PNf‖Lp ,(1.18)

‖P≤Nf‖Lq � N3p− 3

q ‖P≤Nf‖Lp ,(1.19)

‖PNf‖Lq � N3p− 3

q ‖PNf‖Lp .(1.20)

2. Local conservation laws

In this section we record some standard facts about the (non)conservationof mass, momentum and energy densities for general nonlinear Schrodingerequations of the form6

i∂tφ + Δφ = N(2.1)

on the spacetime slab I0×Rd with I0 a compact interval. Our primary interestis of course the quintic defocusing case (1.1) on I0 × R3 when N = |φ|4φ, butwe will also discuss here the U(1)-gauge invariant Hamiltonian case, whenN = F ′(|φ|2)φ with R-valued F . Later on we will consider various truncatedversions of (1.1) with non-Hamiltonian forcing terms. These local conservationlaws will be used not only to imply the usual global conservation of mass andenergy, but also derive “almost conservation” laws for various localized portionsof mass, energy, and momentum, where the localization is either in physicalspace or frequency space. The localized momentum inequalities are closely

6We will use φ to denote general solutions to Schrodinger-type equations, reserving thesymbol u for solutions to the quintic defocusing nonlinear Schrodinger equation (1.1).


related to virial identities, and will be used later to deduce an interactionMorawetz inequality which is crucial to our argument.

To avoid technicalities (and to justify all exchanges of derivatives andintegrals), let us work purely with fields φ, N which are smooth, with allderivatives rapidly decreasing in space; in practice, we can then extend theformulae obtained here to more general situations by limiting arguments. Webegin by introducing some notation which will be used to describe the massand momentum (non)conservation properties of (2.1).

Definition 2.1. Given a (Schwartz) solution φ of (2.1) we define the massdensity

T00(t, x) := |φ(t, x)|2,the momentum density

T0j(t, x) := Tj0(t, x) := 2Im(φφj),

and the (linear part of the) momentum current

Ljk(t, x) = Lkj(t, x) := −∂j∂k|φ(t, x)|2 + 4Re(φjφk).

Definition 2.2. Given any two (Schwartz) functions f, g : Rd → C, wedefine the mass bracket

{f, g}m := Im(fg)(2.2)

and the momentum bracket

{f, g}p := Re(f∇g − g∇f).(2.3)

Thus {f, g}m is a scalar-valued function, while {f, g}p defines a vector field onRd. We will denote the jth component of {f, g}p by {f, g}j

p.

With these notions we can now express the mass and momentum (non)-conservation laws for (2.1), which can be validated with straightforward com-putations.

Lemma 2.3 (Local conservation of mass and momentum). If φ is a(Schwartz ) solution to (2.1) then there exist the local mass conservation iden-tity

∂tT00 + ∂jT0j = 2{N , φ}m(2.4)

and the local momentum conservation identity

∂tT0j + ∂kLkj = 2{N , φ}jp.(2.5)

Here we adopt the usual7 summation conventions for the indices j, k.

7Repeated Euclidean coordinate indices are summed. As the metric is Euclidean, we willnot systematically match subscripts and superscripts.


Observe that the mass current coincides with the momentum density in(2.5), while the momentum current in (2.5) has some “positive definite” ten-dencies (think of Δ = ∂k∂k as a negative definite operator, whereas the ∂j willeventually be dealt with by integration by parts, reversing the sign). These twofacts will underpin the interaction Morawetz estimate obtained in Section 10.

We now specialize to the gauge invariant Hamiltonian case, when N =F ′(|φ|2)φ; note that (1.1) would correspond to the case F (|φ|2) = 1

3 |φ|6. Ob-serve that

{F ′(|φ|2)φ, φ}m = 0(2.6)

and

{F ′(|φ|2)φ, φ}p = −∇G(|φ|2)(2.7)

where G(z) := zF ′(z) − F (z). In particular, for the quintic case (1.1) we have

{|φ|4φ, φ}p = −23∇|φ|6.(2.8)

Thus, in the gauge invariant case we can re-express (2.5) as

∂tT0j + ∂kTjk = 0(2.9)

where

Tjk := Ljk + 2δjkG(|φ|2)(2.10)

is the (linear and nonlinear) momentum current. Integrating (2.4) and (2.9)in space we see that the total mass∫

Rd

T00 dx =∫

Rd

|φ(t, x)|2 dx

and the total momentum∫Rd

T0j dx = 2∫

Rd

Im(φ(t, x)∂jφ(t, x)) dx

are both conserved quantities. In this Hamiltonian setting one can also verifythe local energy conservation law

∂t

[12|∇φ|2 +

12F (|φ|2)

]+ ∂j

[Im(φkφkj) − F ′(|φ|2)Im(φφj)

]= 0(2.11)

which implies conservation of total energy∫Rd

12|∇φ|2 +

12F (|φ|2) dx.

Note also that (2.10) continues the tendency of the right-hand side of (2.5)to be “positive definite”; this is a manifestation of the defocusing nature ofthe equation. Later in our argument, however, we will be forced to deal withfrequency-localized versions of the nonlinear Schrodinger equations, in whichone does not have perfect conservation of mass and momentum, leading to anumber of unpleasant error terms in our analysis.


3. Review of Strichartz theory in R1+3

In this section we review some standard (and some slightly less standard)Strichartz estimates in three dimensions, and their application to the well-posedness and regularity theory for (1.1). We use Lq

tLrx to denote the spacetime

norm

‖u‖Lqt Lr

x(R×R3) :=

(∫R

(∫R3

|u(t, x)|r dx

)q/r

dt

)1/q

,

with the usual modifications when q or r is equal to infinity, or when thedomain R × R3 is replaced by a smaller region of spacetime such as I × R3.When q = r we abbreviate Lq

tLqx as Lq

t,x.

3.1. Linear Strichartz estimates. We say that a pair (q, r) of exponents isadmissible if 2

q + 3r = 3

2 and 2 ≤ q, r ≤ ∞; examples include (q, r) = (∞, 2),(10, 30/13), (5, 30/11), (4, 3), (10/3, 10/3), and (2, 6).

Let I × R3 be a spacetime slab. We define8 the L2 Strichartz normS0(I × R3) by

‖u‖S0(I×R3) := sup(q,r) admissible

( ∑N

‖PNu‖2Lq

t Lrx(I×R3)

)1/2(3.1)

and for k = 1, 2 we then define the Hk Strichartz norm Sk(I × R3) by

‖u‖Sk(I×R3) := ‖∇ku‖S0(I×R3).

We shall work primarily with the H1 Strichartz norm, but will need the L2 andH2 norms to control high frequency and low frequency portions of the solutionu respectively.

We observe the elementary inequality

∥∥∥( ∑N

|fN |2)1/2∥∥∥

Lqt Lr

x(I×R3)≤

( ∑N

‖fN‖2Lq

t Lrx(I×R3)

)1/2(3.2)

for all 2 ≤ q, r ≤ ∞ and arbitrary functions fN ; this is easy to verify inthe extreme cases (q, r) = (2, 2), (2,∞), (∞, 2), (∞,∞), and the intermediatecases then follow by complex interpolation. In particular, (3.2) holds for alladmissible exponents (q, r). From this and the Littlewood-Paley inequality

8The presence of the Littlewood-Paley projections here may seem unusual, but they arenecessary in order to obtain a key L4

t L∞x endpoint Strichartz estimate below.


(see e.g. [40]) we have

‖u‖Lqt Lr

x(I×R3) �∥∥∥( ∑

N

|PNu|2)1/2∥∥∥

Lqt Lr

x(I×R3)

�( ∑

N

‖PNu‖2Lq

t Lrx(I×R3)

)1/2

� ‖u‖S0(I×R3)

and hence

‖∇u‖Lqt Lr

x(I×R3) � ‖u‖S1(I×R3).(3.3)

Indeed, the S1 norm controls the following spacetime norms:

Lemma 3.1 ([44]). For any Schwartz function u on I × R3,

‖∇u‖L∞t L2

x+ ‖∇u‖L10

t L30/13x

+ ‖∇u‖L5tL

30/11x

+ ‖∇u‖L4t L3

x+ ‖∇u‖L

10/3t,x

+ ‖∇u‖L2t L6

x+ ‖u‖L4

t L∞x

+ ‖u‖L6t L18

x+ ‖u‖L10

t,x+ ‖u‖L∞

t L6x

� ‖u‖S1 .(3.4)

where all spacetime norms are on I × R3.

Proof. All of these estimates follow from (3.3) and Sobolev embeddingexcept for the L4

t L∞x norm, which is a little more delicate because endpoint

Sobolev embedding does not work at L∞x . Write

cN := ‖PN∇u‖L2t L6

x+ ‖PN∇u‖L∞

t L2x;

then by the definition of S1 we have( ∑N

c2N

)1/2� ‖u‖S1 .

On the other hand, for any dyadic frequency N we see from Bernstein’s in-equality (1.20) and (1.18) that

N12 ‖PNu‖L2

t L∞x

� cN and N− 12 ‖PNu‖L∞

t L∞x

� cN .

Thus, if aN (t) := ‖PNu(t)‖L∞x

, we have( ∫IaN (t)2 dt

)1/2� N− 1

2 cN(3.5)

and

supt∈I

aN (t) � N12 cN .(3.6)

Let us now compute

‖u‖4L4

t L∞x

�∫

I

( ∑N

aN (t))4

dt.


Expanding this out and using symmetry, we have

‖u‖4L4

tL∞x

�∑

N1≥N2≥N3≥N4

∫IaN1(t)aN2(t)aN3(t)aN4(t) dt.

Estimating the two highest frequencies using (3.5) and the lowest two using(3.6), we can bound this by

�∑

N1≥N2≥N3≥N4

N123 N

124

N121 N

122

cN1cN2cN3cN4 .

Let cN denote the quantity

cN :=∑N ′

min(N/N ′, N ′/N)1/10cN ′ .

Clearly we can bound the previous expression by

�∑

N1≥N2≥N3≥N4

N123 N

124

N121 N

122

cN1 cN2 cN3 cN4 .

But we have cNj� (N1/Nj)1/10cN1 for j = 2, 3, 4; hence we can bound the

above by

�∑

N1≥N2≥N3≥N4

N123 N

124

N121 N

122

c4N1

(N1/N2)1/10(N1/N3)1/10(N1/N4)1/10.

Summing in N4, then in N3, then in N2, we see that this is bounded by∑N1

c4N1

�( ∑

N

c2N

)2.

But by Young’s inequality this is bounded by � (∑N

c2N )2 � ‖u‖4

S1 , and the

claim follows.

We have the following standard Strichartz estimates:

Lemma 3.2. Let I be a compact time interval, and let u : I × R3 → C bea Schwartz solution to the forced Schrodinger equation

iut + Δu =M∑

m=1

Fm

for some Schwartz functions F1, . . . , FM . Then

‖u‖Sk(I×R3) � ‖u(t0)‖Hk(R3) + CM∑

m=1

‖∇kFm‖L

q′mt L

r′mx (I×R3)

(3.7)

for any integer k ≥ 0, any time t0 ∈ I, and any admissible exponents (q1, r1), . . .. . . , (qm, rm), where p′ denotes the dual exponent to p; thus 1/p′ + 1/p = 1.


Proof. We first observe that we may take M = 1, since the claim forgeneral M then follows from the principle of superposition (exploiting thelinearity of the operator (i∂t + Δ), or equivalently using the Duhamel formula(1.13)) and the triangle inequality. We may then take k = 0, since the estimatefor higher k follows simply by applying ∇k to both sides of the equation andnoting that this operator commutes with (i∂t + Δ). The Littlewood-Paleyprojections PN also commute with (i∂t + Δ), and so

(i∂t + Δ)PNu = PNF1

for each N . From the Strichartz estimates in [32] we obtain

‖PNu‖Lqt Lr

x(I×R3) � ‖PNu(t0)‖L2(R3) + ‖PNF1‖L

q′1t L

r′1x (I×R3)

for any admissible exponents (q, r), (q1, r1). Finally, we square, sum this in N

and use the dual of (3.2) to obtain the result.

Remark 3.3. In practice we shall take k = 0, 1, 2 and M = 1, 2, and(qm, rm) to be either (∞, 2) or (2, 6); i.e., we shall measure part of the inho-mogeneity in L1

t Hkx , and the other part in L2

t Wk,6/5x .

3.2. Bilinear Strichartz estimate. It turns out that to control the inter-actions between very high frequency and very low frequency portions of theSchrodinger solution u, Strichartz estimates are insufficient, and we need thefollowing bilinear refinement, which we state in arbitrary dimension (thoughwe need it only in dimension d = 3).

Lemma 3.4. Let d ≥ 2. For any spacetime slab I∗ × Rd, any t0 ∈ I∗, andfor any δ > 0,

‖uv‖L2t L2

x(I∗×Rd) ≤ C(δ)(‖u(t0)‖H−1/2+δ + ‖(i∂t + Δ)u‖L1t H

−1/2+δx

)

× (‖v(t0)‖H

d−12 −δ + ‖(i∂t + Δ)v‖

L1t H

d−12 −δ

x

).(3.8)

This estimate is very useful when u is high frequency and v is low fre-quency, as it moves plenty of derivatives onto the low frequency term. Thisestimate shows in particular that there is little interaction between high andlow frequencies; this heuristic will underlie many of our arguments to come,especially when we begin to control the movement of mass, momentum, andenergy from high modes to low or vice versa. This estimate is essentially therefined Strichartz estimate of Bourgain in [3] (see also [5]). We make the trivialremark that the L2

t,x norm of uv is the same as that of uv, uv, or uv, thus theabove estimate also applies to expressions of the form O(uv).

Proof. We fix δ, and allow our implicit constants to depend on δ. We beginby addressing the homogeneous case, with u(t) := eitΔζ and v(t) := eitΔψ and


consider the more general problem of proving

‖uv‖L2t,x

� ‖ζ‖Hα1‖ψ‖Hα2 .(3.9)

Scaling invariance of this estimate demands that α1 + α2 = d2 − 1. Our first

goal is to prove this for α1 = −12 + δ and α2 = d−1

2 − δ. The estimate (3.9)may be recast using duality and renormalization as

∫g(ξ1 + ξ2, |ξ1|2 + |ξ2|2)|ξ1|−α1 ζ(ξ1)|ξ2|−α2ψ(ξ2)dξ1dξ2 � ‖g‖L2‖ζ‖L2‖ψ‖L2 .

(3.10)

Since α2 ≥ α1, we may restrict attention to the interactions with |ξ1| ≥ |ξ2|.Indeed, in the remaining case we can multiply by ( |ξ2|

|ξ1|)α2−α1 ≥ 1 to return to

the case under consideration. In fact, we may further restrict attention to thecase where |ξ1| > 4|ξ2| since, in the other case, we can move the frequenciesbetween the two factors and reduce to the case where α1 = α2, which canbe treated by L4

t,x Strichartz estimates9 when d ≥ 2. Next, we decompose|ξ1| dyadically and |ξ2| in dyadic multiples of the size of |ξ1| by rewriting thequantity to be controlled as (N, Λ dyadic):∑

N

∑Λ

∫ ∫gN (ξ1 + ξ2, |ξ1|2 + |ξ2|2)|ξ1|−α1 ζN (ξ1)|ξ2|−α2ψΛN (ξ2)dξ1dξ2.

Note that subscripts on g, ζ, ψ have been inserted to evoke the localizations to|ξ1 + ξ2| ∼ N, |ξ1| ∼ N, |ξ2| ∼ ΛN , respectively. Note that in the situation weare considering here, namely |ξ1| ≥ 4|ξ2|, we have that |ξ1 + ξ2| ∼ |ξ1| and thisexplains why g may be so localized.

By renaming components, we may assume that |ξ11 | ∼ |ξ1| and |ξ1

2 | ∼ |ξ2|.Write ξ2 = (ξ1

2 , ξ2). We now change variables by writing u = ξ1+ξ2, v = |ξ1|2+|ξ2|2 and dudv = Jdξ1

2dξ1. A calculation then shows that J = |2(ξ11±ξ1

2)| ∼ |ξ1|.Therefore, upon changing variables in the inner two integrals, we encounter∑

N

N−α1∑Λ≤1

(ΛN)−α2

∫Rd−1

∫R

∫Rd

gN (u, v)HN,Λ(u, v, ξ2)dudvdξ2

where

HN,Λ(u, v, ξ2) =ζN (ξ1)ψΛN (ξ2)

J.

We apply Cauchy-Schwarz on the u, v integration and change back to theoriginal variables to obtain

∑N

N−α1‖gN‖L2

∑Λ≤1

(ΛN)−α2

∫Rd−1

[∫R

∫Rd−1

|ζN (ξ1)|2 |ψΛN (ξ2)|2J

dξ1dξ12

] 12

dξ2.

9In one dimension d = 1, Lemma 3.4 fails when u, v have comparable frequencies, butcontinues to hold when u, v have separated frequencies; see [11] for further discussion.


We recall that J ∼ N and use Cauchy-Schwarz in the ξ2 integration, keepingin mind the localization |ξ2| ∼ ΛN , to get∑

N

N−α1− 12 ‖gN‖L2

∑Λ≤1

(ΛN)−α2+d−12 ‖ζN‖L2‖ψΛN‖L2 .

Choose α1 = −12 + δ and α2 = d−1

2 − δ with δ > 0 to obtain∑N

‖gN‖L2‖ζN‖L2

∑Λ≤1

Λδ‖ψΛN‖L2

which may be summed up to give the claimed homogeneous estimate.We turn our attention to the inhomogeneous estimate (3.8). For simplicity

we set F := (i∂t + Δ)u and G := (i∂t + Δ)v. Then we use Duhamel’s formula(1.13) to write

u = ei(t−t0)Δu(t0)−i

∫ t

t0

ei(t−t′)ΔF (t′) dt′, v = ei(t−t0)Δv(t0)−i

∫ t

t0

ei(t−t′)ΔG(t′).

We obtain10

‖uv‖L2 �∥∥∥ei(t−t0)Δu(t0)ei(t−t0)Δv(t0)

∥∥∥L2

+∥∥∥∥ei(t−t0)Δu(t0)

∫ t

t0

ei(t−t′)ΔG(t′) dt′∥∥∥∥

L2

+∥∥∥∥ei(t−t0)Δv(t0)

∫ t

t0

ei(t−t′)ΔF (t′)dt′∥∥∥∥

L2

+∥∥∥∥∫ t

t0

ei(t−t′)ΔF (t′)dt′∫ t

t0

ei(t−t′′)ΔG(x, t′′) dt′′∥∥∥∥

L2

:= I1 + I2 + I3 + I4.

The first term was treated in the first part of the proof. The second and thethird are similar and so we consider only I2. By the Minkowski inequality,

I2 �∫

R

‖ei(t−t0)Δu(t0)ei(t−t′)ΔG(t′)‖L2 dt′,

and in this case the lemma follows from the homogeneous estimate provedabove. Finally, again by Minkowski’s inequality we have

I4 �∫

R

∫R

‖ei(t−t′)ΔF (t′)ei(t−t′′)ΔG(t′′)‖L2xdt′dt′′,

and the proof follows by inserting in the integrand the homogeneous estimateabove.

10Alternatively, one can absorb the homogeneous components ei(t−t0)Δu(t0), ei(t−t0)Δv(t0)into the inhomogeneous term by adding an artificial forcing term of δ(t − t0)u(t0) andδ(t − t0)v(t0) to F and G respectively, where δ is the Dirac delta.


Remark 3.5. In the situation where the initial data are dyadically lo-calized in frequency space, the estimate (3.9) is valid [3] at the endpointα1 = −1

2 , α2 = d−12 . Bourgain’s argument also establishes the result with α1 =

−12 + δ, α2 = d−1

2 + δ, which is not scale invariant. However, the full estimatefails at the endpoint. This can be seen by calculating the left and right sidesof (3.10) in the situation where ζ1 = χR1 with R1 = {ξ : ξ1 = Ne1 + O(N

12 )}

(where e1 denotes the first coordinate unit vector), ψ2(ξ2) = |ξ2|−d−12 χR2 where

R2 = {ξ2 : 1 � |ξ2| � N12 , ξ2 · e1 = O(1)} and g(u, v) = χR0(u, v) with

R0 = {(u, v) : u = Ne1 + O(N12 ), v = |u|2 + O(N)}. A calculation then shows

that the left side of (3.10) is of size Nd+12 log N while the right side is of size

Nd+12 (log N)

12 . Note that the same counterexample shows that the estimate

‖uv‖L2t,x

� ‖ζ‖Hα1‖ψ‖Hα

2,

where u(t) = eitΔζ, v(t) = eitΔψ, also fails at the endpoint.

3.3. Quintilinear Strichartz estimates. We record the following usefulinequality:

Lemma 3.6. For any k = 0, 1, 2 and any slab I × R3, and any smoothfunctions v1, . . . , v5 on this slab,

(3.11) ‖∇kO(v1v2v3v4v5)‖L1t L2

x

�∑

{a,b,c,d,e}={1,2,3,4,5}‖va‖S1‖vb‖S1‖vc‖L10

x,t‖vd‖L10

x,t||ve||Sk

where all the spacetime norms are on the slab I × R3. In a similar spirit,

‖∇O(v1v2v3v4v5)‖L2t L

6/5x

�5∏

j=1

‖∇vj‖L10t L

30/13x

≤5∏

j=1

‖vj‖S1 .(3.12)

Proof. Consider, for example, the k = 1 case of (3.11). Applying theLeibnitz rule, we encounter various terms to control including

‖O((∇v1)v2v3v4v5)‖L1t L2

x� ‖∇v1‖

L103

t,x

‖v2‖L4tL∞

x‖v3‖L4

t L∞x‖v4‖L10

t,x‖v5‖L10

t,x.

The claim follows then by (3.4). The k = 2 case of (3.11) follows similarly byestimates such as

‖O((∇2v1)v2v3v4v5)‖L1t L2

x� ‖∇2v1‖

L103

t,x

‖v2‖L4t L∞

x‖v3‖L4

t L∞x‖v4‖L10

t,x‖v5‖L10

t,x

and

‖O((∇v1)(∇v2)v3v4v5)‖L1tL2

x� ‖∇v1‖

L103

t,x

‖∇v2‖L4t L∞

x‖v3‖L4

t L∞x‖v4‖L10

t,x‖v5‖L10

t,x.

The k = 0 case is similar (omit all the ∇s).


Finally, estimate (3.12) similarly follows from the Sobolev embedding‖u‖L10

t,x� ‖∇u‖L10

t L30/13x

, (3.4) and Holder’s inequality,

‖O(∇v1v2v3v4v5)‖L2t L

6/5x

� ‖∇v1‖L10t L

30/13x

‖v2‖L10t,x‖v3‖L10

t,x‖v4‖L10

t,x‖v5‖L10

t,x.

We need a variant of the above lemma which also exploits the bilinearStrichartz inequality in Lemma 3.4 to obtain a gain when some of the factorsare “high frequency” and others are “low frequency”.

Lemma 3.7. Suppose vhi, vlo are functions on I × R3 such that

‖vhi‖S0 + ‖(i∂t + Δ)vhi‖L1t L2

x(I×R3) � εK,

‖vhi‖S1 + ‖∇(i∂t + Δ)vhi‖L1t L2

x(I×R3) � K,

‖vlo‖S1 + ‖∇(i∂t + Δ)vlo‖L1t L2

x(I×R3) � K,

‖vlo‖S2 + ‖∇2(i∂t + Δ)vlo‖L1t L2

x(I×R3) � εK

for some constants K > 0 and 0 < ε � 1. Then for any j = 1, 2, 3, 4,

‖∇O(vjhiv

5−jlo )‖L2

tL6/5x (I×R3) � ε

910 K5.

Remark 3.8. The point here is the gain of ε9/10, which cannot be obtaineddirectly from the type of arguments used to prove Lemma 3.6. As the proofwill reveal, one can replace the exponent 9/10 with any exponent less thanone, though for our purposes all that matters is that the power of ε is positive.The S0 bound on vhi effectively restricts vhi to high frequencies (as the lowand medium frequencies will then be very small in S1 norm); similarly, the S2

control on vlo effectively restricts vlo to low frequencies. This lemma is thusan assertion that the components of the nonlinearity in (1.1) arising from in-teractions between low and high frequencies are rather weak; this phenomenonunderlies the important frequency localization result in Proposition 4.3, butthe motif of controlling the interaction between low and high frequencies un-derlies many other parts of our argument also, notably in Proposition 4.9 andProposition 4.15.

Proof. Throughout this proof all spacetime norms shall be on I ×R3. Wemay normalize K := 1. By the Leibnitz rule we have

‖∇O(vjhiv

5−jlo )‖L2

tL6/5x

� ‖O(vjhiv

4−jlo ∇vlo)‖L2

tL6/5x

+ ‖O(vj−1hi v5−j

lo ∇vhi)‖L2t L

6/5x

.

Consider the ∇vlo terms first, which are rather easy. By Holder we have

‖O(vjhiv

4−jlo ∇vlo)‖L2

t L6/5x

� ‖∇vlo‖L∞t L6

x‖vhi‖L∞

t L2x‖vlo‖4−j

L6t L18

x‖vhi‖j−1

L6tL18

x.

Applying (3.4), this is bounded by

� ‖vlo‖S2‖vhi‖S0‖vlo‖4−j

S1‖vhi‖j−1

S1� ε2

which is acceptable.


Now consider the ∇vhi terms, which are more difficult. First consider thej = 2, 3, 4 cases. By Holder we have

‖O(vj−1hi v5−j

lo ∇vhi)‖L2t L

6/5x

� ‖∇vhi‖L2t L6

x‖vlo‖L∞

t L∞x‖vhi‖1/2

L∞t L2

x‖vlo‖4−j

L∞t L6

x‖vhi‖j−3/2

L∞t L6

x.

Now observe (for instance from (1.20), (1.18) and dyadic decomposition) that

‖vlo‖L∞t L∞

x� ‖vlo‖1/2

L∞t L6

x‖∇vlo‖1/2

L∞t L6

x.

Thus, by (3.4),

‖O(vj−1hi v5−j

lo ∇vhi)‖L2t L

6/5x

� ‖vhi‖j−1/2

S1‖vhi‖1/2

S0‖vlo‖1/2

S2‖vlo‖9/2−j

S1

which is O(ε9/10), and is acceptable.Finally consider the j = 1 term. For this term we must use dyadic de-

composition, writing

‖O(v4lo∇vhi)‖L2

t L6/5x

�∑

N1,N2,N3,N4

‖O((PN1vlo)(PN2vlo)(PN3vlo)(PN4vlo)∇vhi)‖L2tL

6/5x

.

By symmetry we may take N1 ≥ N2 ≥ N3 ≥ N4. We then estimate this usingHolder by∑N1≥N2≥N3≥N4

‖O(PN1vlo∇vhi)‖L2t L2

x‖PN2vlo‖L∞

t L6x‖PN3vlo‖L∞

t L6x‖PN4vlo‖L∞

t L∞x

.

The middle two factors can be estimated by ‖vlo‖S1 = O(1). The last factorcan be estimated using Bernstein (1.18) either as

‖PN4vlo‖L∞t L∞

x� N

1/24 ‖PN4vlo‖L∞

t L6x

� N1/24 ‖vlo‖S1 � N

1/24

or as

‖PN4vlo‖L∞t L∞

x� N

−1/24 ‖∇PN4vlo‖L∞

t L6x

� N−1/24 ‖vlo‖S2 � εN

−1/24 .

Meanwhile, the first factor can be estimated using (3.8) as

‖O(PN1vlo∇vhi)‖L2t L2

x� (‖∇vhi(t0)‖H−1/2+δ + ‖(i∂t + Δ)∇vhi‖L1

t H−1/2+δx

)

× (‖PN1vlo(t0)‖H1−δ + ‖(i∂t + Δ)PN1vlo‖L1t H1−δ

x),

where t0 ∈ I is an arbitrary time and 0 < δ < 1/2 is an arbitrary exponent.From the hypotheses on vhi and interpolation we see that

‖∇vhi(t0)‖H−1/2+δ + ‖(i∂t + Δ)∇vhi‖L1t H

−1/2+δx

� ε1/2−δ

while from the hypotheses on vlo and (1.18),

‖PN1vlo(t0)‖H1−δ + ‖(i∂t + Δ)PN1vlo‖L1t H1−δ

x) � N−δ

1 .


Putting this all together, we obtain

‖O(v4lo∇vhi)‖L2

t L6/5x

�∑

N1≥N2≥N3≥N4

ε1/2−δN−δ1 min(N1/2

4 , εN−1/24 ).

Performing the N1 sum, then the N2, then the N3, then the N4, we obtain thedesired bound of O(ε9/10), if δ is sufficiently small.

3.4. Local well-posedness and perturbation theory. It is well known (see e.g.[5]) that the equation (1.1) is locally well-posed11 in H1(R3), and indeed thatthis well-posedness extends to any time interval on which one has a uniformbound on the L10

t,x norm; this can already be seen from Lemma 3.6 and (3.7)(see also Lemma 3.12 below). In this section we detail some variants of thelocal well-posedness argument which describe how we can perturb finite-energysolutions (or near-solutions) to (1.1) in the energy norm when we control theoriginal solution in the L10

t,x norm and the error of near-solutions in a dualStrichartz space. The arguments we give are similar to those in previous worksuch as [5].

We begin with a preliminary result where the near solution, the error ofthe near-solution, and the free evolution of the perturbation are all assumedto be small in spacetime norms, but allowed to be large in energy norm.

Lemma 3.9 (Short-time perturbations). Let I be a compact interval, andlet u be a function on I ×R3 which is a near-solution to (1.1) in the sense that

(i∂t + Δ)u = |u|4u + e(3.13)

for some function e. Suppose that we also have the energy bound

‖u‖L∞t H1

x(I×R3) ≤ E

for some E > 0. Let t0 ∈ I, and let u(t0) be close to u(t0) in the sense that

‖u(t0) − u(t0)‖H1x≤ E′(3.14)

for some E′ > 0. Assume also that there exist the smallness conditions

‖∇u‖L10t L

30/13x (I×R3) ≤ ε0,(3.15)

‖∇ei(t−t0)Δ(u(t0) − u(t0))‖L10t L

30/13x (I×R3) ≤ ε,(3.16)

‖∇e‖L2t L

6/5x

≤ ε(3.17)

for some 0 < ε < ε0, where ε0 is some constant ε0 = ε0(E, E′) > 0.

11In particular, we have uniqueness of this Cauchy problem, at least under the assumptionthat u lies in L10

t,x ∩C0t H1

x, and so whenever we construct a solution u to (1.1) with specifiedinitial data u(t0), we will refer to it as the solution to (1.1) with these data.


Then there exists a solution u to (1.1) on I × R3 with the specified initialdata u(t0) at t0, and furthermore

‖u − u‖S1(I×R3) � E′,(3.18)

‖u‖S1(I×R3) � E′ + E,(3.19)

‖u − u‖L10t,x(I×R3) � ‖∇(u − u)‖L10

t L30/13x (I×R3) � ε,(3.20)

‖∇(i∂t + Δ)(u − u)‖L2t L

6/5x (I×R3) � ε.(3.21)

Note that u(t0) − u(t0) is allowed to have large energy, albeit at the costof forcing ε to be smaller, and worsening the bounds in (3.18). From theStrichartz estimate (3.7), (3.14) we see that the hypothesis (3.16) is redundantif one is willing to take E′ = O(ε).

Proof. By the well-posedness theory reviewed above, it suffices to prove(3.18)–(3.21) as a priori estimates.12 We establish these bounds for t ≥ t0,since the corresponding bounds for the t ≤ t0 portion of I are proved similarly.

First note that the Strichartz estimate (Lemma 3.2), Lemma 3.6 and (3.17)give,

‖u‖S1(I×R3) � E + ‖u‖L10t,x(I×R3) · ‖u‖4

S1(I×R3)+ ε.

By (3.15) and Sobolev embedding we have ‖u‖L10t,x(I×R3) � ε0. A standard

continuity argument in I then gives (if ε0 is sufficiently small depending on E)

‖u‖S1(I×R3) � E.(3.22)

Define v := u − u. For each t ∈ I define the quantity

S(t) := ‖∇(i∂t + Δ)v‖L2tL

6/5x ([t0,t]×R3).

From using Lemma 3.1, Lemma 3.2, (3.16), we have

‖∇v‖L10t L

30/13x ([t0,t]×R3) � ‖∇(v − ei(t−t0)Δv(t0))‖L10

t L30/13x ([t0,t]×R3)(3.23)

+ ‖∇ei(t−t0)Δv(t0)‖L10t L

30/13x ([t0,t]×R3)

� ‖v − ei(t−t0)Δv(t0)‖S1([t0,t]×R3) + ε

� S(t) + ε.(3.24)

On the other hand, since v obeys the equation

(i∂t + Δ)v = |u + v|4(u + v) − |u|4u − e =5∑

j=1

O(vj u5−j) − e

12That is, we may assume the solution u already exists and is smooth on the entire inter-val I.


by (1.15), we easily check using (3.12), (3.15), (3.17), (3.24) that

S(t) � ε +5∑

j=1

(S(t) + ε)jε5−j0 .

If ε0 is sufficiently small, a standard continuity argument then yields the boundS(t) � ε for all t ∈ I. This gives (3.21), and (3.20) follows from (3.24).Applying Lemma 3.2, (3.14) we then conclude (3.18) (if ε is sufficiently small),and then from (3.22) and the triangle inequality we conclude (3.19).

We will actually be more interested in iterating the above lemma13 to dealwith the more general situation of near-solutions with finite but arbitrarilylarge L10

t,x norms.

Lemma 3.10 (Long-time perturbations). Let I be a compact interval, andlet u be a function on I × R3 which obeys the bounds

‖u‖L10t,x(I×R3) ≤ M(3.25)

and

‖u‖L∞t H1

x(I×R3) ≤ E(3.26)

for some M, E > 0. Suppose also that u is a near-solution to (1.1) in the sensethat it solves (3.13) for some e. Let t0 ∈ I, and let u(t0) be close to u(t0) inthe sense that

‖u(t0) − u(t0)‖H1x≤ E′

for some E′ > 0. Assume also the smallness conditions,

‖∇ei(t−t0)Δ(u(t0) − u(t0))‖L10t L

30/13x (I×R3) ≤ ε,(3.27)

‖∇e‖L2t L

6/5x (I×R3) ≤ ε,

for some 0 < ε < ε1, where ε1 is some constant ε1 = ε1(E, E′, M) > 0. Nowthere exists a solution u to (1.1) on I ×R3 with the specified initial data u(t0)at t0, and furthermore

‖u − u‖S1(I×R3) ≤ C(M, E, E′),

‖u‖S1(I×R3) ≤ C(M, E, E′),

‖u − u‖L10t,x(I×R3) ≤ ‖∇(u − u)‖L10

t L30/13x (I×R3) ≤ C(M, E, E′)ε.

Once again, the hypothesis (3.27) is redundant by the Strichartz estimateif one is willing to take E′ = O(ε); however it will be useful in our applications

13We are grateful to Monica Visan for pointing out an incorrect version of Lemma 3.10 ina previous version of this paper, and also in simplifying the proof of Lemma 3.9.


to know that this lemma can tolerate a perturbation which is large in theenergy norm but whose free evolution is small in the L10

t W1,30/13x norm.

This lemma is already useful in the e = 0 case, as it says that one haslocal well-posedness in the energy space whenever the L10

t,x norm is bounded;in fact one has locally Lipschitz dependence on the initial data. For similarperturbative results see [4], [5].

Proof. As in the previous proof, we may assume that t0 is the lower boundof the interval I. Let ε0 = ε0(E, 2E′) be as in Lemma 3.9. (We need to replaceE′ by the slightly larger 2E′ as the H1 norm of u− u is going to grow slightlyin time.)

The first step is to establish a S1 bound on u. Using (3.25) we maysubdivide I into C(M, ε0) time intervals such that the L10

t,x norm of u is atmost ε0 on each such interval. By using (3.26) and Lemmas 3.2, 3.6 as in theproof of (3.22) we see that the S1 norm of u is O(E) on each of these intervals.Summing up over all the intervals we conclude

‖u‖S1(I×R3) ≤ C(M, E, ε0)

and in particular by Lemma 3.1

‖∇u‖L10t L

30/13x (I×R3) ≤ C(M, E, ε0).

We can then subdivide the interval I into N ≤ C(M, E, ε0) subintervals Ij ≡[Tj , Tj+1] so that on each Ij we have,

‖∇u‖L10t L

30/13x (Ij×R3) ≤ ε0.

We can then verify inductively using Lemma 3.9 for each j that if ε1 is suffi-ciently small depending on ε0, N , E, E′, then

‖u − u‖S1(Ij×R3) ≤ C(j)E′,

‖u‖S1(Ij×R3) ≤ C(j)(E′ + E),

‖∇(u − u)‖L10t L

30/13x (Ij×R3) ≤ C(j)ε,

‖∇(i∂t + Δ)(u − u)‖L2t L

6/5x (Ij×R3) ≤ C(j)ε.

Hence by Strichartz (3.7) and (3.4) we have

‖∇ei(t−Tj+1)Δ(u(Tj+1) − u(Tj+1))‖L10t L

30/13x (I×R3)

≤ ‖∇ei(t−Tj)Δ(u(Tj) − u(Tj))‖L10t L

30/13x (I×R3) + C(j)ε

and‖u(Tj+1) − u(Tj+1)‖H1 ≤ ‖u(Tj) − u(Tj)‖H1 + C(j)ε,

allowing one to continue the induction (if ε1 is sufficiently small dependingon E, N , E′, ε0, then the quantity in (3.14) will not exceed 2E′). The claimfollows.


Remark 3.11. The value of ε1 given by the above lemma deteriorates ex-ponentially with M , or more precisely it behaves like exp(−MC) in its de-pendence14 on M . As this lemma is used quite often in our argument, thiswill cause the final bounds in Theorem 1.1 to grow extremely rapidly in E,although they will still of course be finite for each E.

A related result involves persistence of L2 or H2 regularity:

Lemma 3.12 (Persistence of regularity). Let k = 0, 1, 2, I be a compacttime interval, and let u be a finite energy solution to (1.1) on I × R3 obeyingthe bounds

‖u‖L10t,x(I×R3) ≤ M.

Then, if t0 ∈ I and u(t0) ∈ Hk,

‖u‖Sk(I×R3) ≤ C(M, E(u))‖u(t0)‖Hk .(3.28)

In particular, once we control the L10t,x norm of a finite energy solution, we

in fact control all the Strichartz norms in S1, and can even control the S2 normif the initial data are in H2(R3). From this and standard iteration arguments,one can in fact show that a Schwartz solution can be continued in time as longas the L10

t,x norm does not blow up to infinity.

Proof. By the local well-posedness theory it suffices to prove (3.28) as ana priori bound.

Applying Lemma 3.10 with u := u, e := 0, and E′ := 0 we obtain thebound

‖u‖S1(I×R3) � C(M, E).(3.29)

By (3.11) we also have

‖∇kO(u5)‖L1t L2

x� ‖u‖L10

x,t‖u‖Sk‖u‖3

S1 ;(3.30)

the main thing to observe here is the presence of one factor of ‖u‖L10x,t

on theright-hand side.

As in the proof of Lemma 3.10, divide the time interval I into N ≈(1 + M

δ

)10 subintervals Ij := [Tj , Tj + 1] on which

‖u‖L10x,t(Ij×R3) ≤ δ(3.31)

14With respect to all its parameters, ε1(E, E′, M) ≈ exp(−MC〈E〉C〈E′〉C).


where δ will be chosen momentarily. We have on each Ij by the Strichartzestimates (Lemma 3.2) and (3.30),

‖u‖Sk(Ij×R3) ≤ C(‖u(Tj)‖Hk(R3) + ‖∇k(|u|4u)‖L1

t L2x(Ij×R3)

)≤ C

(‖u(Tj)‖Hk(R3) + ‖u‖L10

x,t(Ij×R3) · ‖u‖Sk(I×R3) · ‖u‖3S1(Ij×R3)

).

Choosing δ ≤ (2CC(M, E))−3 gives,

‖u‖Sk(Ij×R3) ≤ 2C‖u(Tj)‖Hk(R3).(3.32)

The bound (3.28) now follows by adding up the bounds (3.32) we have on eachsubinterval.

4. Overview of proof of global spacetime bounds

We now outline the proof of Theorem 1.1, breaking it down into a numberof smaller propositions.

4.1. Zeroth stage: Induction on energy. We say that a solution u to (1.1)is Schwartz on a slab I × R3 if u(t) is a Schwartz function for all t ∈ I; notethat such solutions are then also smooth in time as well as space, thanks to(1.1).

The first observation is that in order to prove Theorem 1.1, it suffices todo so for Schwartz solutions. Indeed, once one obtains a uniform L10

t,x(I × R3)bound for all Schwartz solutions and all compact I, one can then approximatearbitrary finite energy initial data by Schwartz initial data and use Lemma3.10 to show that the corresponding sequence of solutions to (1.1) converges inS1(I × R3) to a finite energy solution to (1.1). We omit the standard details.

For every energy E ≥ 0 we define the quantity 0 ≤ M(E) ≤ +∞ by

M(E) := sup{‖u‖L10t,x(I∗×R3)}

where I∗ ⊂ R ranges over all compact time intervals, and u ranges over allSchwartz solutions to (1.1) on I∗ × R3 with E(u) ≤ E. We shall adopt theconvention that M(E) = 0 for E < 0. By the above discussion, it suffices toshow that M(E) is finite for all E.

In the argument of Bourgain [4] (see also [5]), the finiteness of M(E) inthe spherically symmetric case is obtained by an induction on the energy E;indeed a bound of the form

M(E) ≤ C(E, η, M(E − η4))

is obtained for some explicit 0 < η = η(E) � 1 which does not collapse to 0for any finite E, and this easily implies via induction that M(E) is finite for allE. Our argument will follow a similar induction on energy strategy; howeverit will be convenient to run this induction in the contrapositive, assuming for


contradiction that M(E) can be infinite. We study the minimal energy Ecrit

for which this is true, and then obtaining a contradiction using the “inductionhypothesis” that M(E) is finite for all E < Ecrit. This will be more convenientfor us, especially as we will require more than one small parameter η.

We turn to the details and assume for contradiction that M(E) is notalways finite. From Lemma 3.10 we see that the set {E : M(E) < ∞} is open;clearly it is also connected and contains 0. By our contradiction hypothesis,there must therefore exist a critical energy 0 < Ecrit < ∞ such that M(Ecrit) =+∞, but M(E) < ∞ for all E < Ecrit. One can think of Ecrit as the minimalenergy required to create a blowup solution. For instance, we have

Lemma 4.1 (Induction on energy hypothesis). Let t0 ∈ R, and let v(t0)be a Schwartz function such that E(v(t0)) ≤ Ecrit − η for some η > 0. Thenthere exists a Schwartz global solution v : Rt × R3

x → C to (1.1) with initialdata v(t0) at time t = t0 such that ‖v‖L10

t,x(R×R3) ≤ M(Ecrit − η) = C(η).Furthermore we have ‖v‖S1(R×R3) ≤ C(η).

Indeed, this lemma follows immediately from the definition of Ecrit, thelocal well-posedness theory in L10

t,x, and Lemma 3.12.As in the argument in [4], we will need a small parameter 0 < η =

η(Ecrit) � 1 depending on Ecrit. In fact, our argument is somewhat lengthyand we will actually use seven such parameters

1 η0 η1 η2 η3 η4 η5 η6 > 0.

Specifically, we will need a small quantity 0 < η0 = η0(Ecrit) � 1 assumedto be sufficiently small depending on Ecrit. Then we need a smaller quan-tity 0 < η1 = η1(η0, Ecrit) � 1 assumed sufficiently small depending on Ecrit,η0 (in particular, it may be chosen smaller than positive quantities such asM(Ecrit − η100

0 )−1). We continue in this fashion, choosing each 0 < ηj � 1 tobe sufficiently small depending on all previous quantities η0, . . . , ηj−1 and theenergy Ecrit, all the way down to η6 which is extremely small, much smallerthan any quantity depending on Ecrit, η0, . . . , η5 that will appear in our argu-ment. We will always assume implicitly that each ηj has been chosen to besufficiently small depending on the previous parameters. We will often displaythe dependence of constants on a parameter; e.g. C(η) denotes a large constantdepending on η, and c(η) will denote a small constant depending on η. Whenη1 η2, we will understand c(η1) c(η2) and C(η1) � C(η2).

Since M(Ecrit) is infinite, it is, in particular, larger than 1/η6. By defi-nition of M , this means that we may find a compact interval I∗ ⊂ R and asmooth solution u : I∗ × R3 → C to (1.1) with Ecrit/2 ≤ E(u) ≤ Ecrit so thatu is ridiculously large in the sense that

‖u‖L10t,x(I∗×R3) > 1/η6.(4.1)


We will show that this leads to a contradiction.15 Although u does not actuallyblow up (it is assumed smooth on all of the compact interval I∗), it is stillconvenient to think of u as almost16 blowing up in L10

t,x in the sense of (4.1).We summarize the above discussion with the following:

Definition 4.2. A minimal energy blowup solution of (1.1) is a Schwartzsolution on a time interval I∗ with energy,17

12Ecrit ≤ E(u)(t) =

∫12|∇u(t, x)|2 +

16|u(t, x)|6 dx ≤ Ecrit(4.2)

with the L10x,t norm enormous in the sense of (4.1).

We remark that both conditions (4.1), (4.2) are invariant under the scal-ing (1.3) (though of course the interval I∗ will be dilated by λ2 under thisscaling). Thus applying the scaling (1.3) to a minimal energy blowup solu-tion produces another minimal energy blowup solution. Some of the proofs ofthe sub-propositions below will revolve around a specific frequency N ; usingthis scale invariance, we can then normalize that frequency to equal 1 for theduration of that proof. (Different parts of the argument involve different keyfrequencies, but we will not run into problems because we will only normalizeone frequency at a time).

Henceforth we will not mention the Ecrit dependence of our constantsexplicitly, as all our constants will depend on Ecrit. We shall need however tokeep careful track of the dependence of our argument on η0, . . . , η6. Broadlyspeaking, we will start with the largest η, namely η0, and slowly “retreat” toincreasingly smaller values of η as the argument progresses (such a retreat willfor instance usually be required whenever the induction hypothesis Lemma 4.1is invoked). However we will only retreat as far as η5, not η6, so that (4.1) willeventually lead to a contradiction when we show that

‖u‖L10t,x(I∗×R3) ≤ C(η0, . . . , η5).

Together with our assumption that we are considering a minimal energyblowup solution u as in Definition 4.2, Sobolev embedding implies the bounds

15Assuming, of course, that the parameters η0, . . . , η6 are each chosen to be sufficientlysmall depending on previous parameters. It is important to note however that the ηj cannotbe chosen to be small depending on the interval I∗ or the solution u; our estimates must beuniform with respect to these parameters.

16For instance, u might genuinely blow up at some time T∗ > 0, but I∗ is of the formI∗ = [0, T∗ − ε] for some very small 0 < ε 1, and thus u remains Schwartz on I∗ × R3.

17We could modify our arguments below to allow the assumption here E(u) = Ecrit. Forexample, the arguments in the proof of Proposition 4.3 below also show that the functionM(s) := supE(u)=s{‖u‖L10

x,t} is a nondecreasing function of s. On first reading, the reader

may imagine E(u) = Ecrit in Definition 4.2.


on kinetic energy

‖u‖L∞t H1

x(I∗×R3) ∼ 1(4.3)

and potential energy

‖u‖L∞t L6

x(I∗×R3) � 1(4.4)

(since our implicit constants are allowed to depend on Ecrit). Note that we donot presently have any lower bounds on the potential energy, but see below.

Having displayed our preliminary bounds on the kinetic and potentialenergy, we briefly discuss the mass

∫R3 |u(t, x)|2 dx, which is another conserved

quantity. Because of our a priori assumption that u is Schwartz, we know thatthis mass is finite. However, we cannot obtain uniform control on this massusing our bounded energy assumption, because the very low frequencies of u

may simultaneously have very small energy and very large mass. Furthermoreit is dangerous to rely too much on this conserved mass for this energy-criticalproblem as the mass is not invariant under the natural scaling (1.3) of theequation (indeed, it is super-critical with respect to that scaling). On theother hand, from (4.3) and (1.16) we know that the high frequencies of u havesmall mass:

‖P>Mu‖L2(R3) � 1M

for all M ∈ 2Z.(4.5)

Thus we will still be able to use the concept of mass in our estimates as longas we restrict our attention to sufficiently high frequencies.

4.2. First stage: Localization control on u. We aim to show that a mini-mal energy blowup solution as in Definition 4.2 does not exist. Intuitively, itseems reasonable to expect that a minimal-energy blowup solution should be“irreducible” in the sense that it cannot be decoupled into two or more compo-nents of strictly smaller energy that essentially do not interact with each other(i.e. each component also evolves via (1.1) modulo small errors), since one ofthe components must then also blow up, contradicting the minimal-energy hy-pothesis. In particular, we expect at every time that such a solution should belocalized in both frequency and space.

The first main step in the proof of Theorem 1.1 is to make the aboveheuristics rigorous for our solution u. Roughly speaking, we would like toemphasize that at each time t, the solution u(t) is localized in both space andfrequency to the maximum extent allowable under the uncertainty principle(i.e. if the frequency is localized to N(t), we would like to localize u(t) spatiallyto the scale 1/N(t)).

These sorts of localizations already appear for instance in the argumentof Bourgain [4], [5], where the induction on energy argument is introduced.


Informally,18 the reason that we can expect such localization is as follows.Suppose for contradiction that at some time t0 the solution u(t0) can be splitinto two parts u(t0) = v(t0)+w(t0) which are widely separated in either spaceor frequency, and which each carry a nontrivial amount O(ηC) of energy forsome η5 ≤ η ≤ η0. Then by orthogonality we expect v and w to each havestrictly smaller energy than u, e.g. E(v(t0)), E(w(t0)) ≤ Ecrit − O(ηC). Thusby Lemma 4.1 we can extend v(t) and w(t) to all of I∗ × R3 by evolving thenonlinear Schrodinger equation (1.1) for v and w separately, and furthermorewe have the bounds

‖v‖L10t,x(I∗×R3), ‖w‖L10

t,x(I∗×R3) ≤ M(Ecrit − O(ηC)) ≤ C(η).

Since v and w both solve (1.1) separately, and v and w were assumed to bewidely separated, we thus expect v + w to solve (1.1) approximately. The ideais then to use the perturbation theory from Section 3.4 to obtain a bound of theform ‖u‖L10

t,x(I∗×R3) ≤ C(η), which contradicts (4.1) if η6 is sufficiently small.A model example of this type of strategy occurs in Bourgain’s argument

[5], where substantial effort is invested in locating a “bubble” - a small localizedpocket of energy - which is sufficiently isolated in physical space from the restof the solution. One then removes this bubble, the remainder of the solutionevolves, and then one uses perturbation theory, augmented with the additionalinformation about the isolation of the bubble, to place the bubble back in. Wewill use arguments similar to these in the sequel, but first we need insteadto show that a solution of (1.1) which is sufficiently delocalized in frequencyspace is globally spacetime bounded. More precisely, we have:

Proposition 4.3 (Frequency delocalization implies spacetime bound).Let η > 0, and suppose there exists a dyadic frequency Nlo > 0 and a timet0 ∈ I∗ such that the energy separation conditions hold :

‖P≤Nlou(t0)‖H1(R3) ≥ η(4.6)

and

‖P≥K(η)Nlou(t0)‖H1(R3) ≥ η.(4.7)

If K(η) is sufficiently large depending on η, i.e.

K(η) ≥ C(η)

then,

‖u‖L10t,x(I∗×R3) ≤ C(η).(4.8)

18The heuristic that minimal energy blowup solutions should be strongly localized in bothspace and frequency has been employed in previous literature for a wide variety of nonlinearequations, including many of elliptic or parabolic type. Our formalizations of this heuristic,however, rely on the induction on energy methods of Bourgain and perturbation theory, asopposed to variational or compactness arguments.


We prove this in Section 5. The basic idea is as outlined in previousdiscussion; the main technical tool needed is the multilinear improvements toStrichartz’ inequality in Section 3.3 to control the interaction between the twocomponents and thus allow the resconstruction of the original solution u.

Clearly the conclusion of Proposition 4.3 is in conflict with the hypothesis(4.1), and so we should now expect the solution to be localized in frequencyfor every time t. This is indeed the case:

Corollary 4.4 (Frequency localization of energy at each time). A min-imal energy blowup solution of (1.1) (see Definition 4.2) satisfies: For everytime t ∈ I∗ there exists a dyadic frequency N(t) ∈ 2Z such that for everyη5 ≤ η ≤ η0 we have small energy at frequencies � N(t),

‖P≤c(η)N(t)u(t)‖H1 ≤ η,(4.9)

small energy at frequencies N(t),

‖P≥C(η)N(t)u(t)‖H1 ≤ η,(4.10)

with the large energy at frequencies ∼ N(t),

‖Pc(η)N(t)<·<C(η)N(t)u(t)‖H1 ∼ 1.(4.11)

Here 0 < c(η) � 1 � C(η) < ∞ are quantities depending on η.

Informally, this corollary asserts that at every given time t the solution u

is essentially concentrated at a single frequency N(t). Note however that we donot presently have any information as to how N(t) evolves in time; obtaininglong-term control on N(t) will be a key objective of later stages of the proof.

Proof. For each time t ∈ I∗, we define N(t) as

N(t) := sup{N ∈ 2Z : ‖P≤Nu(t)‖H1 ≤ η0}.

Since u(t) is Schwartz, we see that N(t) is strictly larger than zero; from thelower bound in (4.3) we see that N(t) is finite. By definition of N(t),

‖P≤2N(t)u(t)‖H1 > η0.

Now let η5 ≤ η ≤ η0. Observe that we now have (4.10) if C(η) is chosensufficiently large, because if (4.10) failed then Proposition 4.3 would implythat ‖u‖L10

t,x(I∗×R3) ≤ C(η), contradicting (4.1) if η6 is sufficiently small. Inparticular we have (4.10) for η = η0. Since we also have (4.9) for η = η0 byconstruction of N(t), we thus see from (4.3) that we have (4.11) for η = η0,which of course then implies (again by (4.3)) the same bound for all η5 ≤ η ≤η0. Finally, we obtain (4.9) for all η5 ≤ η ≤ η0 if c(η) is chosen sufficientlysmall, since if (4.9) failed then by combining it with (4.11) and Proposition 4.3we would once again imply that ‖u‖L10

t,x(I∗×R3) ≤ C(η), contradicting (4.1).


Having shown that any minimal energy blowup solution u must be local-ized in frequency at each time, we now turn to showing that such a u is alsolocalized in physical space. This turns out to be somewhat more involved,although it still follows the same general strategy. We first borrow a usefultrick from [4]; since u is Schwartz, we may divide the interval I∗ into threeconsecutive pieces I∗ := I− ∪ I0 ∪ I+ where each of the three intervals containsa third of the L10

t,x density:∫I

∫R3

|u(t, x)|10 dxdt =13

∫I∗

∫R3

|u(t, x)|10 dxdt for I = I−, I0, I+.

In particular from (4.1) we have

‖u‖L10t,x(I×R3) � 1/η6 for I = I−, I0, I+.(4.12)

Thus to contradict (4.1) it suffices to obtain L10t,x bounds on just one of the

three intervals I−, I0, I+.It is in the middle interval I0 that we can obtain physical space localiza-

tion; this will be done in several stages. The first step is to ensure that thepotential energy

∫R3 |u(t, x)|6 dx is bounded from below.

Proposition 4.5 (Potential energy bounded from below). For any min-imal energy blowup solution of (1.1) (see Definition 4.2), for all t ∈ I0,

‖u(t)‖L6x≥ η1.(4.13)

This is proven in Section 6, and is inspired by a similar argument ofBourgain [4]. Using (4.13) and some simple Fourier analysis, we can thusestablish the following concentration result:

Proposition 4.6 (Physical space concentration of energy at each time).Any minimal energy blowup solution of (1.1) satisfies: For every t ∈ I0, thereexists an x(t) ∈ R3 such that∫

|x−x(t)|≤C(η1)/N(t)|∇u(t, x)|2 dx � c(η1)(4.14)

and ∫|x−x(t)|≤C(η1)/N(t)

|u(t, x)|p dx � c(η1)N(t)p

2−3(4.15)

for all 1 < p < ∞, where the implicit constant can depend on p. In particular,we have ∫

|x−x(t)|≤C(η1)/N(t)|u(t, x)|6 dx � c(η1).(4.16)


This is proven in Section 7. Similar results were obtained in [4], [20] in theradial case; see also [3]. Informally, the above estimates emphasize that u(t, x)is roughly of size N(t)1/2 on the average when |x − x(t)| � 1/N(t); observethat this is consistent with bounded energy (4.3) as well as with Corollary 4.4and the uncertainty principle.

It turns out that in our argument, it is not enough to know that theenergy concentrates at one location x(t) at each time; we must also show thatthe energy is small at all other locations, where |x−x(t)| 1/N(t). The maintool for achieving this is

Proposition 4.7 (Physical space localization of energy at each time).For any minimal energy blowup solution of (1.1), for every t ∈ I0∫

|x−x(t)|>1/(η2N(t))|∇u(t, x)|2 dx � η1.(4.17)

This is proven in Section 8. The proof follows a similar strategy to thatused to prove Proposition 4.3; the main difference is that we now consider spa-tially separated components of u rather than frequency separated components,and instead of using multilinear Strichartz estimates to establish the decou-pling of these components, we shall rely instead on approximate finite speedof propagation and on the pseudoconformal identity.

To summarize, at each time t we have a location x(t), around which thekinetic and potential energy are large, and away from which the kinetic energyis small (and one can also show the potential energy is small, although we willnot need this). From this and a little Fourier analysis we obtain an importantconclusion:

Proposition 4.8 (Reverse Sobolev inequality). When u is a minimalenergy blowup solution (and hence (4.2), (4.9)–(4.17) hold), then for everyt0 ∈ I0, any x0 ∈ R3, and any R ≥ 0,∫

B(x0,R)|∇u(t0, x)|2 dx � η1 + C(η1, η2)

∫B(x0,C(η1,η2)R)

|u(t0, x)|6 dx.(4.18)

Thus, up to an error of η1, we are able to control the kinetic energy locallyby the potential energy.19 This will be proved in Section 9. This fact will becrucial in the interaction Morawetz portion of our argument when we have anerror term involving the kinetic energy, and control of a positive term which

19Note that this is a special property of the minimal energy blowup solution, reflectingthe very strong physical space localization properties of such a solution; it is false in general,even for solutions to the free Schrodinger equation. Of course, Proposition 4.5 is similarlyfalse in general; for instance, for solutions of the free Schrodinger equation, the L6

x norm goesto zero as t → ±∞.


involves the potential energy; the reverse Sobolev inequality is then used tocontrol the former by the latter.

To summarize, the statements above tell us that any minimal energyblowup solution (Definition 4.2) to the equation (1.1) must be localized in bothfrequency and physical space at every time. We are still far from done: we havenot yet precluded blowup in finite time (which would happen if N(t) → ∞ ast → T∗ for some finite time T∗), nor have we eliminated soliton or soliton-likesolutions (which would correspond, roughly speaking, to N(t) staying close toconstant for all time t). To achieve this we need spacetime integrability boundson u. Our main tool for this is a frequency-localized version of the interactionMorawetz estimate (1.8), to which we now turn.

4.3. Second stage: Localized Morawetz estimate. In order to localize theinteraction Morawetz inequality, it turns out to be convenient to work at the“minimum” frequency attained by u.

From (1.18) we observe that

‖Pc(η0)N(t)<·<C(η0)N(t)u(t)‖H1 ≤ C(η0)N(t)‖u‖L∞t L2

x.

Comparing this with (4.11) we obtain the lower bound

N(t) ≥ c(η0)‖u‖−1L∞

t L2x

for t ∈ I0. Since u is Schwartz, the right-hand side is nonzero, and thus thequantity

Nmin := inft∈I0

N(t)

is strictly positive.From (4.9) we see that the low frequency portion of the solution - where

|ξ| ≤ c(η0)Nmin - has small energy; one might then hope to use Strichartzestimates to obtain some spacetime control on these low frequencies. However,we do not yet have much control on the high frequencies |ξ| ≥ c(η0)Nmin, apartfrom the energy bounds (4.3) and (4.4) of course.

Our initial spacetime bound in the high frequencies is provided by thefollowing interaction Morawetz estimate.

Proposition 4.9 (Frequency-localized interaction Morawetz estimate).When u is a minimal energy blowup solution of (1.1) (and hence (4.2), (4.9)–(4.18) all hold), then for all N∗ < c(η3)Nmin∫

I0

∫|P≥N∗u(t, x)|4 dxdt � η1N

−3∗ .(4.19)

Remark 4.10. The factor N−3∗ on the right-hand side of (4.19) is man-

dated by scale-invariance considerations (cf. (1.3)). The η1 factor on the rightside reflects our smallness assumption on N∗: if we think of N∗ as being very


small and then scale the solution so that N∗ = 1, we are pushing the energyto very high frequencies so heuristically it’s not unreasonable to expect thesupercritical L4

x,t norm on the left-hand side to be small.Regarding the size of N∗: write for the moment c(η3) as the constant

appearing in Corollary 4.4 with η = η3. The constant c(η3) appearing inProposition 4.9 is chosen so that c(η3) � c(η3) ·η3. Hence at all times we knowthere is very little energy at frequencies below N∗

η3, and (ignoring factors of

N∗ which can be scaled to 1) above frequency N∗ there is very little (at mostη3/N∗) L2 mass.

This small η1 factor will be used to close a bootstrap argument in the proofof the important estimate on the movement of energy to very low frequenciesin Lemma 15.1 below.

If one already had Theorem 1.1, then this proposition would follow (butwith η1 replaced by C(Ecrit)) from Lemma 3.12, since the S1 norm will control‖∇u‖L4

t L3x

and hence ‖|∇|3/4u‖L4t L4

xby Sobolev embedding. Of course, we

will not prove Proposition 4.9 this way, as it would be circular. Instead, thisproposition is based on the interaction Morawetz inequality developed in [12],[13] (see also a recent extension in [24]). The key thing about this estimate isthat the right-hand side does not depend on I0; thus for instance it is alreadyuseful in eliminating soliton or pseudosoliton solutions, at least for frequenciesclose to Nmin. (Frequencies much larger than Nmin still cause difficulty, andwill be dealt with later in the argument). Proposition 4.9 roughly correspondsto the localized Morawetz inequality used by Bourgain [4], [5] and Grillakis[20] in the radial case (see (1.7) above). The main advantage of (4.19) is thatit is not localized to near the spatial origin, in contrast with the standard (1.5)and localized (1.7) Morawetz inequalities.

Although this proposition is based on the interaction Morawetz inequalitydeveloped in the references given above, there are significant technical difficul-ties in truncating that inequality to the high frequencies. As a consequence theproof of this proposition is somewhat involved and is given in Sections 10-14.Also, we caution the reader that the above proposition is not proved as an apriori estimate; indeed the proof relies crucially on the assumption that u is aminimal energy blowup solution in the sense of (4.1), and in particular verifiesthe reverse Sobolev inequality (4.18). See Section 10 for further remarks onthe proof.

Combining Proposition 4.9 with Proposition 4.6 gives us the followingintegral bound on N(t).

Corollary 4.11. For any minimal energy blowup solution of (1.1),∫I0

N(t)−1 dt � C(η1, η3)N−3min.(4.20)


Proof. Let N∗ := c(η3)Nmin for some sufficiently small c(η3). Then fromProposition 4.9,∫

I0

∫R3

|P≥N∗u(t, x)|4 dxdt � η1N−3∗ � C(η1, η3)N−3

min.

On the other hand, from Bernstein (1.19) and (4.4), for each t ∈ I0 that∫|x−x(t)|≤C(η1)/N(t)

|P<N∗u(t, x)|4 dx�N(t)−3‖P<N∗u(t)‖4L∞

x�C(η1)N(t)−3N2

∗ .

So by (4.15) and the triangle inequality (since N∗ ≤ c(η3)N(t)),∫R3

|P≥N∗u(t, x)|4 dx � c(η1)N(t)−1.

Comparing this with the previous estimate, the claim follows.

Remark 4.12. The estimate (4.20) is scale-invariant under the naturalscaling (1.3) (N has the units of length−1, and t has the units of length2).In the radial case, a somewhat similar estimate was obtained by Bourgain [4]and implicitly also by Grillakis [20]; in our notation, this bound would be theassertion that ∫

IN(t) dt � |I|1/2(4.21)

for all I ⊆ I0; indeed in the radial case (when x(t) = 0) this bound easily followsfrom Proposition 4.6 and (1.7). Both estimates are equally good at estimatingthe amount of time for which N(t) is comparable to Nmin, but Corollary 4.11is much weaker than (4.21) when it comes to controlling the times for whichN(t) Nmin. Indeed if we could extend (4.21) to the nonradial case we couldobtain a significantly shorter proof of Theorem 1.1. However we were unableto prove this bound directly,20 although it can be deduced from Corollary 4.11and Proposition 4.15 below).

This Corollary allows us to obtain some useful L10t,x bounds in the case

when N(t) is bounded from above.

Corollary 4.13 (Nonconcentration implies spacetime bound). Let I ⊆I0, and suppose there exists an Nmax > 0 such that N(t) ≤ Nmax for all t ∈ I.

20Comparing (4.19) with (1.7) one also sees that our control on how often the solutionconcentrates is weaker than that in the radial arguments of Bourgain and Grillakis. Heuristi-

cally: (4.19) allows a long train in time of N3 “bubbles” at frequency N � 1, with size ∼ N12 ,

spatial extent N−1, and individual duration ∼ N−2, so the total lifespan of the train is ∼ N .The estimate (1.7), on the other hand, restricts such bubbles to a set of dimension less than12

in time. Our proof of Theorem 1.1 makes up for this weakness in its next stage, specificallythe relatively strong frequency localized L2 almost conservation estimate of Lemma (4.14).


Then for any localized minimal energy blowup solution of (1.1),

‖u‖L10t,x(I×R3) � C(η1, η3, Nmax/Nmin)

and furthermore

‖u‖S1(I×R3) � C(η1, η3, Nmax/Nmin).

Proof. We may use scale invariance (1.3) to rescale Nmin = 1. FromCorollary 4.11 we obtain the useful bound

|I| � C(η1, η3, Nmax).

Let δ = δ(η0, Nmax) > 0 be a small number to be chosen later. We maypartition I into O(|I|/δ) intervals I1, . . . , IJ of length at most δ. Let Ij be anyof these intervals, and let tj be any time in Ij . Observe from Corollary 4.4 andthe hypothesis N(tj) ≤ Nmax that

‖P≥C(η0)Nmaxu(tj)‖H1 ≤ η0

(for instance). Now let u(t) := ei(t−tj)ΔP<C(η0)Nmaxu(tj) be the free evolution

of the low and medium frequencies of u. The above estimate then becomes

‖u(tj) − u(tj)‖H1 ≤ η0.

On the other hand, from Bernstein (1.19) and (4.3) we have

‖∇u(t)‖L30/13x

� C(η0, Nmax)‖u(tj)‖H1 � C(η0, Nmax)

for all t ∈ Ij , and hence

‖∇u‖L10t L

30/13x (Ij×R3) � C(η0, Nmax)δ1/10.

Similarly,

‖∇(|u(t)|4u(t))‖L6/5x

� ‖∇u(t)‖L6x‖u(t)‖4

L6x

�C(η0, Nmax)‖u(tj)‖5H1 � C(η0, Nmax)

and hence‖∇(|u(t)|4u(t))‖L2

t L6/5x (Ij×R3) � C(η0, Nmax)δ1/2.

From these two estimates, the energy bound (4.3), and Lemma 3.9 with e =−|u|4u, we see (if δ is chosen sufficiently small) that

‖u‖L10t,x(Ij×R3) � 1

Summing this over each of the O(|I|/δ) intervals Ij we obtain the desired L10t,x

bound. The S1 bound then follows from Lemma 3.12.


The corollary above gives the desired contradiction to (4.12) whenNmax/Nmin is bounded; i.e., N(t) stays in a bounded range.

4.4. Third stage: Nonconcentration of energy. Of course, any globalwell-posedness argument for (1.1) must eventually exclude a blowup scenario(self-similar or otherwise) where N(t) goes to infinity in finite time, and indeedby Corollary 4.13 this is the only remaining possibility for a minimal energyblowup solution. Corollary 4.4 implies that in such a scenario the energy mustalmost entirely evacuate the frequencies near Nmin, and instead concentrateat frequencies much larger than Nmin. While this scenario is consistent withconservation of energy, it turns out to not be consistent with the time andfrequency distribution of mass.

More specifically, we know there is a tmin ∈ I0 so that for all t ∈ I0, N(t) ≥N(tmin) := Nmin > 0. By Corollary 4.4, at time tmin the solution has the bulkof its energy near the frequency Nmin, and hence the medium frequencies atthat time have mass bounded below by,

‖Pc(η0)Nmin≤·≤C(η0)Nminu(tmin)‖L2 � c(η0)N−1

min.(4.22)

The idea is to prove the following approximate mass conservation law for thesehigh frequencies,21 which states that while some mass might slip to very lowfrequencies as the solution moves to high frequencies, it cannot all do so.

Lemma 4.14 (Some mass freezes away from low frequencies). Suppose u

is a minimal energy blowup solution of (1.1), and let [tmin, tevac] ⊂ I0 be suchthat N(tmin) = Nmin and N(tevac)/Nmin ≥ C(η5). Then for all t ∈ [tmin, tevac],

‖P≥η1004 Nmin

u(t)‖L2 � η1N−1min.(4.23)

Lemma 4.14 will quickly show that the evacuation scenario - wherein thesolution cleanly concentrates energy to very high frequencies - cannot occur.Instead the solution always leaves a nontrivial amount of mass and energybehind at medium frequencies. This “littering” of the solution will serve (viaCorollary 4.4) to keep N(t) from escaping to infinity22 and gives us,

21It is necessary to truncate to the high frequencies in order to exploit mass conserva-tion because the low frequencies contain an unbounded amount of mass. This strategy ofmollifying the solution in frequency space in order to exploit a conservation law that wouldotherwise be unbounded or useless is inspired by the “I-method” for sub-critical dispersiveequations; see e.g. [13].

22It is interesting to note that one must exploit conservation of energy, conservation ofmass, and conservation of momentum (via the Morawetz inequality) in order to preventblowup for the equation (1.1); the same phenomenon occurs in the previous arguments [4],[20] in the radial case, even though the details of those arguments are in many ways quitedifferent from those here.


Proposition 4.15 (Energy cannot evacuate from low frequencies). Forany minimal energy blowup solution of (1.1),

N(t) � C(η5)Nmin(4.24)

for all t ∈ I0.

We give the somewhat complicated proof of Lemma 4.14 and Proposition4.15 in Section 15.

By combining Proposition 4.15 with Corollary 4.13, we encounter a con-tradiction to (4.12) which completes the proof of Theorem 1.1.

The proofs of the above claims occupy the remainder of this paper. Beforemoving to these proofs, we summarize the role of the parameters ηi, i = 0, . . . , 5which have now all been introduced. The number η1 represents the amountof potential energy that must be present at every time in a minimal energyblowup solution (Proposition 4.5); it also represents the extent of concentrationof energy (on the scale of 1/N(t)) that must occur in physical space at everytime in a minimal energy blowup solution (Proposition 4.6). The number η2

is introduced in Proposition 4.7, where 1/η2 represents the extent that thereis localization (on the scale of 1/N(t)) of energy in a minimal energy blowupsolution. The number η3 measures, on the scale of the quantity Nmin, whatwe mean by “high frequency” when we say Proposition 4.9 is an interactionMorawetz estimate localized to high frequencies. The number η4 measuresthe frequency (on the scale of Nmin) below which the evolution can’t move acertain portion (namely, η1) of the L2 mass. Finally, the number η0 enters inCorollary 4.4 and various other points in the paper where we simply use itsvalue as a small, universal constant.

5. Frequency delocalized at one time =⇒ spacetime bounded

We now prove Proposition 4.3. Let 0 < ε = ε(η) � 1 be a small number tobe chosen later. If K(η) is sufficiently large depending on ε, then one can findat least ε−2 disjoint intervals [ε2Nj , Nj/ε2], j = 1, . . . , ε−2, contained inside[Nlo, K(η)Nlo]. From (4.3) and the pigeonhole principle, there must thereforeexist an Nj so that the interval [ε2Nj , Nj/ε2] is mostly free of energy:

‖Pε2Nj≤·≤Nj/ε2u(t0)‖H1 � ε.(5.1)

As the statement and conclusion of Proposition 4.3 is invariant under thescaling (1.3) we may set Nj := 1. Now define

ulo(t0) := P≤εu(t0); uhi(t0) := P≥1/εu(t0).

The functions ulo(t0), uhi(t0) have strictly smaller energy than u:


Lemma 5.1. If ε is sufficiently small depending on η, then

E(ulo(t0)), E(uhi(t0)) ≤ Ecrit − cηC .

Proof. We prove this for E(ulo(t0)); the claim for E(uhi(t0)) is similar.Define uhi′(t0) := P>εu(t0), so that u(t0) = ulo(t0) + uhi′(t0), and consider thequantity

|E(u(t0)) − E(ulo(t0)) − E(uhi′(t0))|.(5.2)

From (1.2) we can bound this by

(5.2) � |〈∇ulo(t0),∇uhi′(t0)〉| + |∫

|u(t0)|6 − |ulo(t0)|6 − |uhi′(t0)|6 dx|.(5.3)

The functions ulo and uhi′ almost have disjoint supports, and their inner prod-uct is very close to zero. Indeed from Parseval and (5.1) we have

|〈∇ulo(t0),∇uhi′(t0)〉| � ε2.

Now we estimate the L6-type terms. From the pointwise estimate∣∣|u(t0)|6 − |ulo(t0)|6 − |uhi′(t0)|6∣∣ � |ulo||uhi′ |(|ulo| + |uhi′ |)4

(cf. (1.14)) and Holder’s inequality, we can bound the second term in (5.3) by

� ‖ulo‖∞‖uhi′‖3(‖ulo‖6 + ‖uhi′‖6)4.

From (4.3), (5.1), and Bernstein’s inequality (1.20) we see that

‖ulo‖∞ �∑N≤ε

‖PNu‖∞ �∑N≤ε

N1/2‖PNu‖H1

�∑

N≤ε2

N1/2 +∑

ε2<N≤ε

N1/2ε � ε

and

‖uhi′‖3 �∑N≥ε

‖PNu‖3 �∑N≥ε

N−1/2‖PNu‖H1

�∑

N≥1/ε

N−1/2 +∑

ε≤N≤ 1ε

N−1/2ε � ε1/2.

Thus from (4.4) we can bound the second term in (5.3) by O(ε3/2). Combiningthis with the estimate obtained on the first piece of (5.3), we thus see that

|E(u) − E(ulo(t0)) − E(uhi′(t0))| � ε3/2.

On the other hand, by hypothesis on u we have E(u) ≤ Ecrit, while from (4.7)and (1.2) we have E(uhi′(t0)) � ηC . The claim follows if ε is chosen sufficientlysmall.


From Lemma 5.1 and Lemma 4.1 we know that there exist Schwartz so-lutions ulo, uhi on the slab I∗ ×R3 with initial data ulo(t0), uhi(t0) at time t0,and furthermore

‖ulo‖S1(I∗×R3), ‖uhi‖S1(I∗×R3) � C(η).(5.4)

Let u := ulo + uhi. We now claim that u is an approximate solution to (1.1):

Lemma 5.2. We have

iut + Δu = |u|4u − e

where the error e obeys the bounds

‖∇e‖L2t L

6/5x (I∗×R3) � C(η)ε1/2.(5.5)

Proof. We begin by establishing further estimates on ulo and uhi, beyond(5.4). For uhi, we observe from (4.5) that ‖uhi(t0)‖2 � ε; so from Lemma 3.12,

‖uhi‖S0(I∗×R3) � C(η)ε.(5.6)

Similarly, from (4.3) and (1.17) we have ‖ulo(t0)‖H2 � Cε, and so from Lemma3.12 again we have

‖ulo(t0)‖S2(I∗×R3) � C(η)ε.(5.7)

From Lemma 3.6 we also see that

‖|uhi|4uhi‖L1t L2

x(I∗×R3) � C(η)ε,

‖∇(|uhi|4uhi)‖L1t L2

x(I∗×R3) � C(η),

‖∇(|ulo|4ulo)‖L1t L2

x(I∗×R3) � C(η),

‖∇2(|ulo|4ulo)‖L1t L2

x(I∗×R3) � C(η)ε.

From Lemma 3.7 we thus have

‖∇O(ujhiu

5−jlo )‖L2

t L6/5x (I∗×R3) � C(η)ε1/2

for j = 1, 2, 3, 4. Since e =∑4

j=1 O(ujhiu

5−jlo ) by (1.15), the claim follows.

We now pass from estimates on u to estimates on u by perturbation theory.From (5.1) we have the perturbation bound

‖u(t0) − u(t0)‖H1 � ε,

while from (5.4) we have

‖u‖L10t,x(I∗×R3) � C(η).

Thus if ε is small enough depending on η, we may apply Lemma 3.10 andobtain the desired bound (4.8). The proof of Proposition 4.3 is now complete.


Remark 5.3. The dependence of constants in Proposition 4.3 given by theabove argument is quite poor. Specifically, the separation K(η) needs to be aslarge as

K(η) ≥ C exp(Cη−CM(Ecrit − ηC)C)

(mainly in order for the pigeonhole argument to work) and the bound oneobtains on the L10

t,x norm at the end has a similar size. This implies that thedependence of the constants C(ηj), c(ηj) in Corollary 4.4 is similarly similarly:

C(ηj) ≥ C exp(Cη−Cj M(Ecrit − ηC

j )C),

c(ηj) ≤ (C(ηj))−1.

This will force us to select each ηj+1 quite small depending on previous ηj ;indeed in some cases the induction hypothesis is used more than once andso ηj+1 is even smaller than the above expressions suggest. If one then runsthe induction of energy argument in a direct way (rather than arguing bycontradiction as we do here), this leads to very rapidly growing (but still finite)bound for M(E) for each E, which can only be expressed in terms of multiplyiterated towers of exponentials (the Ackermann hierarchy). More precisely, ifwe use X ↑ Y to denote exponentiation XY ,

X ↑↑ Y := X ↑ (X ↑ . . . ↑ X)

to denote the tower formed by exponentiating Y copies of X,

X ↑↑↑ Y := X ↑↑ (X ↑↑ . . . ↑↑ X)

to denote the double tower formed by tower-exponentiating Y copies of X,and so forth, then we have computed our final bound for M(E) for large E toessentially be

M(E) ≤ C ↑↑↑↑↑↑↑↑ (CEC).

This rather Bunyanesque bound is mainly due to the large number of timeswe invoke the induction hypothesis Lemma 4.1, and is presumably not bestpossible. For instance, the best bound known23 in the radial case is M(E) ≤C ↑↑ (CEC), where the induction hypothesis is used only once; see [4]. Finally,in the case of the subcritical cubic nonlinear Schrodinger equation, the boundfor the analogue of M(E) is polynomial, M(E) ≤ CEC ; see [13].

6. Small L6x norm at one time =⇒ spacetime bounded

We now prove Proposition 4.5. The argument here is similar to the induc-tion on energy arguments in [4]. The point is that the linear evolution of the

23Note added in proof: a bound of M(E) ≤ C ↑ (CEC) in the radial case was recentlyobtained in [45].


solution must concentrate at some point (t1, x1) in spacetime (otherwise wecould iterate using the small data theory). If the solution does not concentratein L6 at time t = t0, then t1 must be far away from t0. The idea is then toremove the energy concentrating at (t1, x1) and induct on energy.

We turn to the details. Assume for contradiction that (4.13) failed forsome time t0 ∈ I0, so that

‖u(t0)‖L6x≤ η1.(6.1)

Fix this t0. By rescaling using (1.3) we may normalize N(t0) = 1. Observethat if the linear solution ei(t−t0)Δu(t0) had small L10

t,x norm, then the standardsmall data well-posedness theory (based on Strichartz estimates and (3.11) or(3.12)) would already show that the nonlinear solution u had bounded L10

norm. Thus we may assume that

‖ei(t−t0)Δu(t0)‖L10t,x(R×R3) � 1.

On the other hand, by Corollary 4.4 we have

‖Plou(t0)‖H1x

+ ‖Phiu(t0)‖H1x

� η0

where we define Plo := P<c(η0) and Phi := P>C(η0); so by Strichartz (Lemma3.2)

‖ei(t−t0)ΔPlou(t0)‖L10t,x(R×R3) + ‖ei(t−t0)ΔPhiu(t0)‖L10

t,x(R×R3) � η0.

If we then define Pmed := 1 − Plo − Phi, we must then have

‖ei(t−t0)ΔPmedu(t0)‖L10t,x(R×R3) ∼ 1.

On the other hand, by (4.3) we know that Pmedu(t0) has bounded energy andhas Fourier support in the region c(η0) � |ξ| � C(η0). Thus by Strichartz (3.7)we have that

‖ei(t−t0)ΔPmedu(t0)‖L10/3t,x (R×R3) � C(η0)

(for instance). From these two estimates and Holder we see that the L∞t,x norm

cannot be too small:

‖ei(t−t0)ΔPmedu(t0)‖L∞t,x(R×R3) � c(η0).

In particular, there exist a time t1 ∈ R and a point x1 such that we have theconcentration

|ei(t1−t0)Δ(Pmedu(t0))(x1)| � c(η0).

By perturbing t1 a little we may assume that t1 �= t0; by time reversal symmetrywe may take t1 < t0.

Let δx1 be the Dirac mass at x1, and let f(t1) := Pmedδx1 . We extend f

to all of R × R3 by the free evolution, thus f(t) := ei(t−t1)Δf(t1). We recordsome explicit estimates on f :


Lemma 6.1. For any t ∈ R and any 1 ≤ p ≤ ∞,

‖f(t)‖Lpx

� C(η0)(1 + |t − t1|)3/p−3/2.

Proof. We may translate t1 = x1 = 0. From the unitarity of eitΔ andBernstein (1.20) we have

‖f(t)‖L∞x (R3) � C(η0)‖f(t)‖L2

x(R3) = C(η0)‖Pmedδx1‖L2x(R3) � C(η0)

while from the dispersive inequality (1.12) we have

‖f(t)‖L∞x (R3) � |t|−3/2‖Pmedδx1‖L1

x(R3) � |t|−3/2.

Combining these estimates we obtain the lemma in the case p = ∞. To obtainthe other cases we need some decay on f(t, x) in the region |x| C(η0)(1+|t|).For this we use the Fourier representation (1.10) to write

f(t, x) =∫

R3

e2πi(x·ξ−2πt|ξ|2)ϕmed(ξ) dξ

where ϕmed is the Fourier multiplier corresponding to Pmed. When |x| 1+|t|,then the phase oscillates in ξ with a gradient comparable in magnitude to |x|.By repeated integration by parts (see e.g. [41]) we thus obtain a bound of theform |f(t, x)| � |x|−100 in this region. Combining this with the previous L∞

bounds we obtain the result.

In particular, from (6.1) and Holder we have

|〈u(t0), f(t0)〉| � η1‖f(t0)‖L6/5x (R3) � C(η0)η1(1 + |t0 − t1|).

On the other hand, we have

|〈u(t0), f(t0)〉| = |〈ei(t1−t0)ΔPmedu(t0), δx1〉| � c(η0).

Thus the concentration point t1 must be far away from t0 (recall that η1 ismuch smaller than η0):

|t1 − t0| � c(η0)η−11 .

In particular, the smallness of η1 pushes the concentration time far away fromthe time when L6

x is small. Since t0 > t1 by hypothesis, we thus see fromLemma 6.1 and the frequency localization of f that ∇f has small L10

t L30/13x

norm to the future of t0:

‖∇f‖L10t L

30/13x ([t0,+∞)×R3) � C(η0)‖f‖L10

t L30/13([t0,+∞)×R3)

� C(η0)‖(1 + |t − t1|)−2/10‖L10t ([t0,+∞))

� C(η0)|t0 − t1|−1/10

� C(η0)η1/101 .

(6.2)

We now use the induction hypothesis (inspired by a similar argument in [4]).We split u(t0) := v(t0)+w(t0) where w(t0) := δeiθΔ−1f(t0), and δ = δ(η0) > 0


is a small number to be chosen shortly, and θ is a phase to be chosen shortly.We now claim that if δ and θ are chosen correctly, then v(t0) has slightlysmaller energy than u. Indeed, we have by integration by parts and definitionof f that

12

∫R3

|∇v(t0)|2 =12

∫R3

|∇u(t0) −∇w(t0)|2

=12

∫R3

|∇u(t0)|2

− δRe∫

R3

e−iθ∇Δ−1f(t0) · ∇u(t0) + O(δ2‖Δ−1f(t0)‖2H1)

≤ Ecrit + δRee−iθ〈u(t0), f(t0)〉 + O(δ2C(η0)).

Since |〈u(t0), f(t0)〉| was already shown to have magnitude at least c(η0), wesee if δ = δ(η0) and θ are chosen correctly that we can ensure that

12

∫R3

|∇v(t0)|2 ≤ Ecrit − c(η0).

Meanwhile, another application of Lemma 6.1 shows that

‖w(t0)‖L6x

� C(η0)‖f(t0)‖L6x

� C(η0)〈t0 − t1〉−1 � C(η0)η1.

So by (6.1) and the triangle inequality,∫R3

|v(t0)|6 � C(η0)η61.

Thus if η1 is sufficiently small depending on η0, then

E(v(t0)) ≤ Ecrit − c(η0).

By Lemma 4.1 we may extend v(t0) into a solution of the nonlinear Schrodingerequation (1.1) on [t0,+∞) such that

‖v‖L10t,x([t0,+∞)×R3) � M(Ecrit − c(η0)) = C(η0).(6.3)

On the other hand, from (6.2) and the frequency localization we have

‖∇ei(t−t0)Δw(t0)‖L10t L

30/13x ([t0,+∞)×R3) � C(η0)η

1/101 .

Thus if η1 is sufficiently small depending on η0 then we may use Lemma 3.10(with u := v and e = 0) to conclude that u extends to all of [t0,+∞) with

‖u‖L10t,x([t0,+∞)×R3) � C(η0, η1).

Since the time interval [t0,+∞) contains I+, this contradicts (4.12). (If we hadt0 < t1 instead, we would obtain a similar contradiction involving I−.) Thisconcludes the proof of Proposition 4.5.


7. Spatial concentration of energy at every time

We now prove Proposition 4.6. Fix t. By scaling using (1.3) we may takeN(t) = 1. By Corollary 4.4 this implies that

‖P>C(η1)u(t)‖H1 + ‖P<c(η1)u(t)‖H1 � η1001(7.1)

(for instance). In particular by Sobolev we have

‖P>C(η1)u(t)‖L6x

+ ‖P<c(η1)u(t)‖L6x

� η1001

and hence by (4.13)‖Pmedu(t)‖L6

x� η1

where Pmed := Pc(η1)<·<C(η1). On the other hand, from (4.3),

‖Pmedu(t)‖L2x

� C(η1)

and thus by Holder’s inequality,

‖Pmedu(t)‖L∞x

� c(η1).

Thus there exists x(t) ∈ R3 such that

c(η1) � |Pmedu(t, x(t))|.(7.2)

Let Kmed denote the kernel associated to the operator Pmed∇Δ−1, and letR > 0 be a radius to be chosen later. Then (7.2) can be continued with

c(η1) � |Kmed ∗ ∇u(t, x(t))|

�∫

|Kmed(x(t) − x)||∇u(t, x)|dx

∼

∫|x−x(t)|<R)


+∫|x−x(t)|≥R


� C(η1)

(∫|x−x(t)|<R

|∇u(t, x)|2dx

) 12

+ C(η1)

(∫|x−x(t)|≥R

|∇u(t, x)||x − x(t)|100 dx

).

Here we used Cauchy-Schwarz and the fact that Kmed is a Schwartz function.Using the fact that (

∫|∇u|2dx)1/2 is bounded uniformly, we obtain

c(η1) �(∫

|x−x(t)|<R|∇u(t, x)|2dx

) 12

+ C(η1)R−10


(say), which proves (4.14) after setting R := C(η1) sufficiently large. Similarly,writing Kmed for the kernel associated to Pmed, we have for all 1 < p < ∞,

c(η1) �∫|x−x(t)|<R

|Kmed(x(t) − x)||u(t, x)|dx

+∫|x−x(t)|≥R

|Kmed(x(t) − x)||u(t, x)|dx

� C(η1)

(∫|x−x(t)|≤R

|u(t, x)|pdx

) 1p

+∫|x−x(t)|≥R

|u(t, x)||x − x(t)|100

dx

� C(η1)

(∫|x−x(t)|≤R

|u(t, x)|pdx

) 1p

+

(∫|x−x(t)|>R

1

|x − x(t)|100× 65

dx

) 56

‖u(t)‖L6x

� C(η1)

(∫|x−x(t)|≤R

|u(t, x)|pdx

) 1p

+ C(η1)R−10,

where we used (4.4). This proves (4.15) upon rescaling, if the radius R = C(η1)was chosen sufficiently large.

8. Spatial delocalized at one time =⇒ spacetime bounded

We now prove Proposition 4.7. This is the spatial analogue of the fre-quency delocalization result in Proposition 4.3. The role of the bilinearStrichartz estimate in that proposition will be played here by finite speedof propagation and pseudoconformal identity estimates. We follow the samebasic strategy as in Proposition 4.3 (but now played out in the arena of phys-ical space rather than frequency space). More specifically, we assume thatthere is a large amount of energy away from the concentration point, and thenuse approximate finite speed of propagation to decouple the solution into twonearly noninteracting components of strictly smaller energy, which can thenbe handled by the induction hypothesis and perturbation theory.

We turn to the details. We first need a number of large quantities. Specif-ically, we need a large integer J = J(η1) 1 to be chosen later, and thena large frequency24 N0 = N0(η1, J) 1 to be chosen later, and then a largeradius R0 = R0(η1, N0, J) 1 to be chosen later.

24More precisely, this is a ratio of two frequencies, but as we will shortly normalize N(0)= 1, the distinction between a frequency and a frequency ratio becomes irrelevant. Similarlythe radii given below should really be ratios of radii.


Suppose for contradiction that the proposition is false. Then there mustexist a time t0 ∈ I0 such that∫

|x−x(t0)|>1/(η2N(t0))|∇u(t0, x)|2 dx � η1.

Here x(t) is the quantity constructed in Proposition 4.6 (see (7.2)). We maynormalize t0 = x(t0) = 0, and rescale so that N(0) = 1; thus∫

|x|>1/η2

|∇u(0, x)|2 dx � η1.(8.1)

On the other hand, if R0 = R0(η1) is chosen large enough then we see fromProposition 4.6 that ∫

|x|<R0

|∇u(0, x)|2 dx � c(η1)(8.2)

and ∫|x|<R0

|u(0, x)|6 dx � c(η1).(8.3)

We then define the radii R0 � R1 � . . . � RJ recursively by Rj+1 := 100R100j .

The region R0 < |x| < RJ can be partitioned into J dyadic shells of theform Rj < |x| < Rj+1. By (4.3),(4.4) and the pigeonhole principle we mayfind 0 ≤ j < J such that∫

Rj<|x|<Rj+1

|∇u(0, x)|2 + |u(0, x)|6 dx � 1J

.(8.4)

Fixing this j, we now introduce cutoff functions χinner, χouter, where χinner isadapted to the ball B(0, 2Rj) and equals one on B(0, Rj), whereas χouter is abump function adapted to B(0, Rj+1) which equals one on B(0, Rj+1/2). Wethen define v(0), w(0) as

v(0, x) := P1/N0≤·≤N0(χinneru(0)); w(0) := P1/N0≤·≤N0

((1 − χouter)u(0)).

(8.5)

By (8.4) we easily see that

‖P1/N0≤·≤N0u(0) − v(0) − w(0)‖H1 � ‖(χouter − χinner)u(0)‖H1 � 1

J1/2.

Also, if N0 is chosen sufficiently large depending on J we see from the normal-ization N(0) = 1 and Corollary 4.4 that

‖u(0) − P1/N0≤·≤N0u(0)‖H1 � 1

J1/2

and thus

‖u(0) − v(0) − w(0)‖H1 � 1J1/2

.(8.6)

We also know that v, w have slightly smaller energy than u:


Lemma 8.1. For the the functions v, w defined in (8.5) we have

E(v(0)), E(w(0)) ≤ Ecrit − c(η1).

Proof. The argument here is completely analogous to that in Lemma5.1, except that now we work in physical space, exploiting the fact that v ismostly supported in the region |x| < 3Rj and w is mostly supported in theregion |x| > Rj+1/2.

We begin by estimating the quantity

|E(v(0) + w(0)) − E(v(0)) − E(w(0))| .

Expanding out the definition of energy, we can bound this by

�∫

R3

|∇v(0, x)||∇w(0, x)| + |v(0, x)||w(0, x)|5 + |v(0, x)|5|w(0, x)| dx.

We subdivide R3 into the regions |x| ≤ Rj+1/2 and |x| > Rj+1/2. Since v

and w are bounded in H1 and hence in L6, we can use Holder’s inequality toestimate the previous expression by

� ‖∇v‖L2(|x|>Rj+1/2)+‖v‖L6(|x|>Rj+1/2)+‖∇w‖L2(|x|≤Rj+1/2)+‖w‖L6(|x|≤Rj+1/2).

Consider for instance the quantity ‖∇v‖L2(|x|>Rj+1/2). Let K be the convolu-tion kernel associated with P1/N0≤·≤N0

; then we have ∇v = (∇(χinneru(0)))∗K.Since χinner is supported on the region |x| ≤ 2Rj ≤ Rj+1/4, we may restrictK to the region |x| > Rj+1/4. On this region, K decays rapidly and in facthas an L1 norm of at most C(N0)/R100

j+1 ≤ C(N0)/R1000 (for instance). Since

∇(χinneru(0)) is bounded in L2, we thus have

‖∇v‖L2(|x|>Rj+1/2) ≤ C(N0)/R1000 .

The other three terms above can be estimated similarly. Thus

|E(v(0) + w(0)) − E(v(0)) − E(w(0))| ≤ C(N0)/R1000 .

On the other hand, from (8.6), the boundedness of u(0), v(0), w(0) in H1 andL6, and Holder’s inequality, we have

|E(u(0)) − E(v(0) + w(0))| � J−1/2.

Thus we have

|E(u(0)) − E(v(0)) − E(w(0))| � J−1/2 + C(N0)/R1000 .(8.7)

On the other hand, by (8.2) and (8.1) (choosing η2 to be smaller than 1/RJ),we know that E(v)(0), E(w)(0) � c(η1). Together with (8.7), this yields thelemma when J = J(η1) and R0 = R0(η1, N0, J) are chosen sufficiently large.


From the above lemma and Lemma 4.1 we may then extend v and w bythe nonlinear Schrodinger equation (1.1) to all of R × R3, so that

‖v‖L10t,x

+ ‖w‖L10t,x

� M(Ecrit − c(η1)) = C(η1).(8.8)

From this, Lemma 3.12 and the frequency localization of v, w we thus obtainthe Strichartz bounds

‖v‖Sk + ‖w‖Sk � C(η1, N0)(8.9)

for k = 0, 1, 2.The idea is now to use our perturbation lemma to approximate u by v+w.

To do this we need to ensure that v and w do not interact. This is the objectiveof the next two lemmas.

Lemma 8.2. Let v(x, t), w(x, t) be the evolutions according to (1.1) of thefunctions defined in (8.5). For times |t| ≤ R10

j , there exists the “finite speed ofpropagation” estimate∫

|x|�R50j

|v(t, x)|2 dx � C(η1, N0)R−20j(8.10)

and for times |t| ≥ R10j , the decay estimate∫

|v(t, x)|6 dx � R−10j .(8.11)

(The powers of Rj are far from sharp.) Meanwhile, the mass density of w

obeys the finite speed of propagation estimate∫|x|�R50

j

|w(t, x)|2 dx � C(η1, N0)R−20j(8.12)

for all |t| ≤ R10j and, similarly, the energy density of w obeys the finite speed

of propagation estimate∫|x|�R50

j

[12|∇w(t, x)|2 +

16|w(t, x)|6]dx � R−20

j C(η1, N0),(8.13)

for all |t| < R10j . Also,∫

|x|�R50j

[12|∇v(t, x)|2 +

16|v(t, x)|6]dx � R−20

j C(η1, N0).(8.14)

Thus at short times t = O(R10j ), v and w are separated in space, whereas

at long times v has decayed (while w is still bounded in Strichartz norms).


Proof. The estimates (8.10) and (8.11) follow from the pseudoconformallaw following arguments from [5]. Recall the pseudoconformal conservationlaw for sufficiently regular and decaying solutions of (1.1):

‖(x + 2it∇)u(t)‖2L2

x+

43t2‖u(t)‖6

L6x

= ‖|x|u0‖2L2

x− 16

3

∫ t

0s‖u(s)‖6

L6xds.(8.15)

Thus, since v solves (1.1),∫|x|�R50

j

|x|2|v(t, x)|2dx � t2‖∇v(t)‖2L2

x+t2‖v(t)‖6

L6x+‖|x|v0‖2

L2x+

∫ t

0s‖v(s)‖6

L6xds.

We restrict to times |t| ≤ R10j and have

R100j

∫|x|�R50

j

|v(t, x)|2dx

� R20j ‖∇v(t)‖2

L∞|t|≤R10

j

L2x

+ R20j ‖v(t)‖6

L∞|t|≤R10

j

L6x

+ R2j‖v0‖L2

x

2

� R20j

(‖∇v(t)‖2

L∞|t|≤R10

j

L2x

+ ‖v(t)‖6L∞

|t|≤R10j

L6x

)+ R2

jN20 ‖∇v0‖2

L2x

� C(N0)R20j E(u0),

which proves (8.10).From (8.15), we observe that

‖v(t)‖6L6

x�

R2jN

20 E(u0)t2

,

so that, for times |t| > R10j , we obtain (8.11).

We control the L2x-mass of w in the ball |x| < 1000R50

j using a virialidentity. Let ζ denote a nonnegative smooth bump function equaling 1 onB(0, 1000R50

j ) and supported on B(0, 2000R50j ). Note that ζ has been chosen

so that the support of ∇ζ does not intersect the support of χinner(0) or thesupport of (1 − χouter)(0). From (2.4), (2.6) and integration by parts we havethe identity

∂t

∫ζ(x)|w(t, x)|2dx = −2

∫ζj(x)Im(wwj)(t, x)dx.

Thus,

|∂t

∫ζ(x)|w(t, x)|2dx| � R−50

j ‖∇w(t)‖L2x‖w(t)‖L2

x,

and we have, using the support properties and (4.3),

sup|t|<R10

j

∫ζ(x)|w(t, x)|2dx �

∫ζ(x)|w(0, x)|2dx

+ R−40j sup

|t|<R10j

‖∇w(t)‖L2x‖w(t)‖L2

x

� R−40j + R−40

j (E(u0))2N0

(say), which proves (8.12).


We now control the energy density of w,

e(w)(t, x) :=12|∇w(t, x)|2 +

16|w(t, x)|6,(8.16)

on the ball |x| < R50j by a similar argument. From (2.11) and integration by

parts we have

d

dt

∫ζe(w)dx =

∫ζj [Im(wkwkj) − δjk|w|4Im(wwk)]dx,

which implies that∫ζe(w)(T )dx �

∫ζe(w)(0)dx +

∫ T

0

∫|∇ζ||∇w||∇∇w|dxdt

+∫ T

0

∫|∇ζ||w|5|∇w|dxdt.

We will control the three terms on the right to obtain (8.13). The first termvanishes due to support properties of ζ and 1 − χouter. The second term iscrudely bounded using (4.3) by

R−50j R10

j ‖∇2w‖L∞|t|<R10

j

L2x.

By the induction hypothesis we have (8.9) and, in particular,

‖∇2w‖L∞t L2

x� C(η1, N0).

The third term is bounded using Holder by, say

� R−50j ‖w‖2

L10t,x‖w‖3

L6t,x‖∇w‖L

10/3t,x

.

Again, the global L10t,x bound and Lemma 3.12 gave us (8.9) which, by inter-

polation, controls all the norms appearing here. This proves (8.13).Replacing ζ by 1−ζ and w by v in the discussion just completed establishes

(8.14).

Corollary 8.3. For v, w as in Lemma 8.2,

‖∇(|v + w|4(v + w) − |v|4v − |w|4w)‖L2tL

6/5x (R×R3)

� C(η1, N0)R−5/6j � C(η1, N0)R

−5/60 .

Proof. By (1.15), the task is to control terms of the form O(vjw4−j∇w)and O(wjv4−j∇v), for j = 1, 2, 3, 4, in L2

t L6/5x . Separate the analysis into

three spacetime regions based on the estimates in Lemma 8.2: (short time,near origin) |t| < R10

j , |x| < 2000R50j ; (short time, far from origin) |t| <

R10j , |x| ≥ 2000R50

j ; (long time) |t| ≥ R10j . In all but one of these cases, an

application of a variant of (3.12),

‖∇u1u2u3u4u5‖L2t L

6/5x

� ‖∇u1‖L∞t L2

x‖u2‖L4

tL∞x‖u3‖L4

t L∞x‖u4‖L∞

t L6x‖u5‖L∞

t L6x,


together with (8.9) and the decay properties in Lemma 8.2 establishes theclaimed estimate controlling the interaction of v and w. The term O(w4∇v)in the long time regime |t| ≥ R10

j presents an exceptional case since we donot directly encounter the available long time decay estimate (8.11). Thissituation is treated separately with the following argument. By Holder andinterpolation, we have

‖O(w4∇v)‖L2t L

6/5x

� ‖∇v‖L∞t L3

x‖O(w4)‖L2

t L2x

� ‖v‖1/2L∞

t L6x‖∇2v‖1/2

L∞t L2

x‖O(w4)‖L2

t L2x

� ‖v‖1/2L∞

t L6x‖v‖1/2

S2‖O(w4)‖L2

t L2x.

Note the appearance of (8.11) which contributes the decay R−5/6j . We complete

the proof by estimating,

‖O(w4)‖L2tL2

x� ‖O(w3)‖L2

t L6x‖w‖L∞

t L3x

� ‖w‖3L6

t L18x‖w‖1/2

L∞t L2

x‖w‖1/2

L∞t L6

x� ‖w‖3

S1‖w(0)‖1/2L2

xE1/4

� C(η1, η2)N1/20 � C(η1, η2, N0).

In light of this corollary, (8.6), (8.8), and the observation that u, v, w

all have bounded energy, we see from Lemma 3.10 (with u := v + w ande := |v + w|4(v + w) − |v|4v − |w|4w) that if J is sufficiently large dependingon η1, and R0 sufficiently large depending on η1, J , N0, then

‖u‖L10t,x(I∗×R3) � C(η1),

which contradicts (4.1). This proves Proposition 4.7.

9. Reverse Sobolev inequality

We now prove Proposition 4.8. Fix t0, x0, R. We may normalize x(t0) = 0and N(t0) = 1. Then by Proposition 4.7 we have∫

|x|>1/η2

|∇u(t0, x)|2 dx � η1.(9.1)

Now suppose for contradiction that we have∫B(x0,R)

|∇u(t0, x)|2 dx η1 + K(η1, η2)∫

B(x0,K(η1,η2)R)|u(t0, x)|6 dx(9.2)

for some large K(η1, η2) to be chosen later. From (9.2) and (9.1) we see thatB(x0, R) cannot be completely contained inside the region |x| > 1/η2; thus,

|x0| � R + 1/η2.(9.3)


Next, we obtain a lower bound on R. Recall from the normalization N(t0) = 1and Corollary 4.4 that

‖P>C(η1)u(t0)‖H1 � η1.

On the other hand, from (9.2),∫B(x0,R)

|∇u(t0, x)|2 dx η1

and from the triangle inequality we see that∫B(x0,R)

|∇P≤C(η1)u(t0, x)|2 dx η1.

But by Holder, Bernstein (1.20), and (4.3) we can bound the left-hand side by

� R3‖∇P≤C(η1)u(t0)‖2L∞

x� C(η1)R3

and thus we have

R � c(η1).

Combining this with (9.3) we see that the ball B(x0, K(η1, η2)R) will con-tain B(0, 1/η2) (and hence any ball of the form B(0, C(η1))) if the constantK(η1, η2) is large enough. In particular, from Proposition 4.6 we have∫

B(x0,K(η1,η2)R)|u(t0, x)|6 dx � c(η1),

which inserted into (9.2) contradicts the energy bound (4.3) if K(η1, η2) ischosen sufficiently large. This proves Proposition 4.8.

10. Interaction Morawetz: generalities

We shall shortly begin the proof of Proposition 4.9, which is a variantof the interaction Morawetz inequality (1.8). As noted above, this inequalitycannot be applied directly to our situation because the right-hand side of (1.8)can be very large due to low frequency contributions to u. It is then natural(in light of (4.5)) to try to adapt the interaction Morawetz inequality to onlydeal with the high frequencies u≥1, but this turns out to not quite be enougheither. The trouble is that the inhomogeneous Schrodinger equation satisfiedby u≥1 is not Lagrangian - and in particular it no longer enjoys the usualL2 conservation. Hence when we apply the argument from [13], [12] (whichgave (1.8)) to this u≥1 equation, we get new terms arising from the fact thatthe right side of (2.4) is no longer zero. We can find no appropriate boundsfor these new terms. Our solution to this problem is to localize the previous


interaction Morawetz arguments in space25, yielding a much more complicatedversion (Theorem 11.1) of the inequality (1.8).

To summarize: the increased complexity on the right-hand side of (11.6)below is due to the fact that we have localized the argument in frequency(because (1.1) is critical) and space (because of the error terms introduced bythe frequency localization). Of course, all of these extra terms will somehowhave to be shown to be bounded, and to this end the second term on theleft side of (11.6) is very important. An analogue of this term - where thex integration is taken over all of R3 - can also be included on the left side of(1.8) (see [13], [12]), but we had previously found no use for this term. In whatfollows, the second term on the left side of (11.6) will be used to absorb — viathe reverse Sobolev inequality of Proposition 4.8 and an averaging argument— some of the most troublesome terms involving kinetic energy that appearon the right side of (11.6).26

The above argument will be carried out in the next section; in this sectionwe prepare for our work involving the non-Lagrangian equation satisfied byu≥1 by discussing interaction Morawetz inequalities in more general situationsthan the quintic NLS (1.1). In particular, we shall consider general solutionsφ to the equation (2.1), where N is an arbitrary nonlinearity.

10.1. Virial-type identity. We introduce two related quantities whichaverage the mass and momentum densities (see Definition 2.1) against a weightfunction a(x).

Definition 10.1. Let a(x) be a function27 defined on the spacetime slabI0 × R3. We define the associated virial potential

Va(t) =∫

R3

a(x)|φ(t, x)|2dx(10.1)

and the associated Morawetz action

Ma(t) =∫

R3

aj2Im(φφj)dx.(10.2)

A calculation using Lemma 2.3 shows that

∂tVa = Ma + 2∫

R3

a{N , φ}mdx,(10.3)

25See (10.5), where the novelty over the arguments from [13], [12] is now the presence ofthe spatial cut-off function.

26The discussion here gives another way to frame our regularity argument which wassketched in Section 4. We only bother to show that a minimal energy blowup solutionmust be localized in space in order that we can apply the reverse Sobolev inequality to suchsolutions. The reverse Sobolev inequality is needed in the proof of the frequency localizedL4

x,t bound.27In other contexts it’s useful to consider also time dependent weight functions a(t, x).


so that Ma = ∂tVa when N = F ′(|φ|2)φ. Using Lemma 2.3, a longer butsimilar calculation establishes,

Lemma 10.2 (Virial-type Identity). Let φ be a (Schwartz ) solution of(2.1). Then

∂tMa =∫

R3

(−ΔΔa)|φ|2 + 4ajkRe(φjφk) + 2aj{N , φ}jpdx.(10.4)

We now infer a useful identity by choosing the weight function a(x) aboveto be,

a(x) = |x|χ(|x|),(10.5)

where χ(r) denotes a smooth nonnegative bump function defined on r ≥ 0,supported on 0 ≤ r ≤ 2 and satisfying χ(r) = 1 for 0 ≤ r ≤ 1. We calculate,

aj(x) =xj

|x| χ(|x|) where χ(r) = χ(r) + rχ′(r),

ajk(x) =1|x|

(δjk − xj

|x|xk

|x|

)χ(|x|) +

xj

|x|xk

|x| χ′(|x|),

Δa(x) =2|x| χ(|x|) + χ′(|x|),

ΔΔa(x) = 2Δ(

1|x|

)χ(|x|) + ψ(|x|),

where ψ(|x|) is smooth and supported in 1 ≤ |x| ≤ 2. Define now the notationM0 := Ma when a(x) is chosen as in (10.5). (Later we will localize around adifferent fixed point y ∈ R3, in which case we’ll write the Morawetz action asMy. Note that the letter a is dropped completely now from the notation forthe Morawetz action.) By the definition (10.2),

M0(t) = 2Im∫

R3

xj

|x| χ(|x|)φj(t, x)φ(t, x)dx.(10.6)

Note that28

|M0(t)| ≤ 2‖φ(t)‖L2x‖φ(t)‖H1 .(10.7)

Inserting the calculation below (10.5) above into (10.4), and using the notation∂r(0) to denote the radial part of the gradient with center at 0 ∈ R3 (so that

28One can also show that |M0(t)| � ‖φ(t)‖2H1/2 ; see Lemma 2.1 in [13].


∂r(0) ≡ x|x| · ∇), we get

∂tM0 = −2

∫R3

Δ(1|x|)|φ(x)|2χ(|x|)dx

+ 4∫

R3

[|∇φ(x)|2 − |∂r(0)φ(x)|2] 1|x| χ(|x|)dx

+ 2∫

R3

(x)|x| · {N , φ}pχ(|x|)dx

−∫

R3

|φ(x)|2ψ(|x|)dx + 4∫

R3

|∂r(0)φ|2χ′(|x|)dx.

As remarked above, we translate the origin and choose, for fixed y ∈ R3

a(x) = |x − y|χ(|x − y|),(10.8)

instead of (10.5), in which case the Morawetz action (10.2) is written My.Then the preceding formula adjusts to give the following spatially localizedvirial-type identity, where we write χ for another bump function with thesame properties as χ,

∂tMy =(10.9)

− 2∫

R3

Δ(1

|x − y|)|φ(x)|2χ(|x − y|)dx(10.10)

+ 4∫

R3

|∇/ yφ(x)|2 1|x − y| χ(|x − y|)dx(10.11)

+ 2∫

R3

(x − y)|x − y| · {N , φ}pχ(|x − y|)dx(10.12)

+ O

(∫R3

(|φ(x)|2 + |∂r(y)φ(x)|2)|ψ(x − y)|dx

).(10.13)

We have used here the notation ∂r(y) to denote the radial portion of the gradientcentered at y and ∇/ y for the rest of the gradient. That is,

∂r(y) :=x − y

|x − y| · ∇ and ∇/ y := ∇− x − y

|x − y|(x − y

|x − y| · ∇).

We have also taken the liberty to dismiss some of the structure in (10.13) usingthe fact that χ′ has the same support properties as ψ.

10.2. Interaction virial identity and general interaction Morawetz estimatefor general equations. When we choose a(x) = |x|χ(x) above, the virial poten-tial reads Va(t) =

∫R3 |φ(x, t)|2|x|χ(x)dx and hence M0(t) := d

dtVa(t) might bethought of as measuring the extent to which the mass of φ (near the origin atleast) is moving away from the origin at time t. Similarly, for fixed y ∈ R3,My(t) gives some measure of the mass movement away from the point y.

Since we are ultimately interested in global decay and scattering propertiesof φ, it’s reasonable to look for some measure of how the mass is moving away


from (or interacting with) itself. We might therefore sum over all y ∈ R3 theextent to which mass is moving away from y (that is, My(t)) multiplied by theamount of mass present at that point y (that is, |φ(y, t)|2dy). The result is thefollowing quantity which we’ll call the spatially localized Morawetz interactionpotential.

M interact(t) =∫

R3y

|φ(t, y)|2My(t)dy(10.14)

= 2Im∫

R3y

∫R3

x

|φ(t, y)|2χ(|x − y|)(x − y)|x − y|(10.15)

· [∇φ(t, x)]φ(t, x)dxdy.

Note that, using (10.7),

|M interact(t)| � ‖φ(t)‖3L2

x‖φ(t)‖H1

x.(10.16)

We calculate, using (10.9) and (2.4),

∂tMinteract =

∫R3

y

|φ(y)|2∂tMydy(10.17)

+∫

R3y

[2∂ykIm(φφk)(y) + 2{N , φ}m]My(t)dy.(10.18)

The ∂yk appearing in (10.18) will now be integrated by parts. Thus, usingLemma 2.3 and the fact that on R3,Δ 1

|x| = −4πδ, we have our spatially local-ized interaction virial-type identity

∂tMinteract =(10.19)

8π

∫R3

|φ(t, y)|4dy(10.20)

+ 4∫

R3y

∫R3

x

|φ(t, y)|2[

1|x − y| χ(|x − y|)

]|∇/ yφ(t, x)|2dxdy(10.21)

+ 2∫

R3y

∫R3

x

|φ(t, y)|2[χ(|x − y|)(x − y)

|x − y|

]· {N , φ}p(t, x)dxdy(10.22)

− 4∫

R3y

∫R3

x

Im(φφk)(t, y)∂yk

[(x − y)j

|x − y| χ(|x − y|)]

Im(φjφ(t, x))dxdy(10.23)

+ O

(∫R3

y

∫R3

x

|φ(t, y)|2|ψ(|x − y|)|[|φ(t, x)|2 + |∂r(y)φ(t, x)|2]dxdy

)(10.24)

+ 4∫

R3y

∫R3

x

{N , φ}m(t, y)[χ(|x − y|)(x − y)

|x − y|

]· Im(φ∇φ)(t, x)dxdy.(10.25)

The ∂yk differentiation in (10.23) produces two terms. When ∂yk falls onχ(|x − y|), we encounter a term controlled by (10.24). Indeed, we get a term


bounded by∫R3

y

∫R3

x

|φ(x)||∂r(y)φ(x)||φ(y)||∂r(x)φ(y)|χ′(|x − y|)dxdy.

Upon grouping the terms in the integrand as [|φ(x)||∂r(x)φ(y)|][|φ(y)||∂r(y)φ(x)|]and using |ab| � |a|2 + |b|2, we find the second part of the expression (10.24).When the derivative falls on the unit vector, we encounter a term of indetermi-nate sign but which is bounded from below by (10.21). We present the details(which also appear in Proposition 2.2 of [13]). The term we are considering is(with t dependence supressed)

− 4∫

R3y

∫R3

x

Im(φφk)(y)[−δjk +

(x − y)j(x − y)k

|x − y|2]

· Im(φjφ)(x)[

1|x − y| χ(|x − y|)

]dxdy

≥ −4∫

R3x

∫R3

y

∣∣∣Im(φ∇/ xφ)(y) · Im(φ∇/ yφ)(x)∣∣∣ [

1|x − y| χ(|x − y|)

]dxdy

≥ −4∫

R3x

∫R3

y

|φ(y)||∇/ xφ(y)||φ(x)||∇/ yφ(x)|[

1|x − y| χ(|x − y|)

]dxdy

≥ −2∫

R3y

∫R3

x

(|φ(y)|2|∇/ yφ(x)|2 + |φ(x)|2|∇/ xφ(y)|2)[

1|x − y| χ(|x − y|)

]dxdy

≥ (10.21).

Thus, apart from another term of the form (10.24), we have shown that−(10.21) and (10.23) together contribute a nonnegative term.

Restricting the above calculations to the time interval I0, we have thefollowing useful estimate.

Proposition 10.3 (Spatially Localized Interaction Morawetz Inequality).Let φ be a (Schwartz ) solution to the equation (2.1) on a spacetime slab I0×R3

for some compact interval I0. Then

8π

∫I0

∫R3

y

|φ(t, y)|4dydt

+ 2∫

I0

∫R3

y

∫R3

x

|φ(t, y)|2[χ(|x − y|)(x − y)

|x − y|

]· {N , φ}p(t, x)dxdydt

≤2‖φ‖3L∞

t L2x(I0×R3)‖φ‖L∞

t H1x(I0×R3)

+ 4∫

I0

∫R3

y

∫R3

x

|{N, φ}m(t, y)||χ(|x − y|)||φ(t, x)||∇φ(t, x)|dxdydt

+ O

(∫I0

∫R3

y

∫R3

x

|φ(t, y)|2|ψ(|x − y|)|[|φ(t, x)|2 + |∂r(y)φ(t, x)|2]dxdydt

).


The proof follows directly by integrating (10.19) over the time interval I0

using (10.16).

Remark 10.4. If we replace (10.8) by

a(x) = |x − y|χ( |x − y|

R

)(10.26)

then there are adjustments to the inequality obtained in Proposition 10.3. Ofcourse, χ(·) is replaced by χ(·/R). The annular cutoff ψ in the final termarises in the analysis above when derivatives fall on χ or χ. A review of thederivation shows that this term adjusts under (10.26) into

O

(∫I0

∫R3

y

∫R3

x

|φ(t, y)|2∣∣∣∣ψ( |x − y|

R

)∣∣∣∣ [1

R3|φ(t, x)|2+

1R|∂r(y)φ(t, x)|2

]dxdydt

).

(10.27)

If we send R → ∞ and specialize to solutions of (1.1), then we can apply (2.6),(2.8) to obtain the bound∫

I0

∫R3

y

|u(t, y)|4dydt

+∫

I0

∫R3

y

∫R3

x

|u(t, y)|2|u(t, x)|6|x − y| dxdydt � ‖u‖3

L∞t L2

x(I0×R3)‖u‖L∞t H1

x(I0×R3),

which is basically (1.8) (see Footnote 10.1). However for the purposes of prov-ing Proposition 4.9, it turns out not to be feasible to send R → ∞, as one ofthe error terms (specifically, the portion of the mass bracket {N , φ}m whichlooks schematically like u5

hiulo) becomes too difficult to estimate.

11. Interaction Morawetz: The setup and an averaging argument

Having discussed interaction Morawetz inequalities in general, we are nowready to begin the proof of Proposition 4.9.

From the invariance of this proposition under the scaling (1.3) we maynormalize N∗ = 1. Since we are assuming 1 = N∗ < c(η3)Nmin, we have inparticular that 1 < c(η3)N(t) for all t ∈ I0. From Corollary 4.4 and Sobolevwe have the low frequency estimate

‖u<1/η3‖L∞

t H1x(I0×R3) + ‖u<1/η3

‖L∞t L6

x(I0×R3) � η3(11.1)

(for instance), if c(η3) was chosen sufficiently small. By (4.11) we thus seethat our choice of N∗ = 1 has forced Nmin ≥ c(η0)η−1

3 . Define Phi := P≥1 andPlo := P<1, and then define uhi := Phiu and ulo := Plou. Observe from (11.1)that ulo has small energy,

‖ulo‖L∞t H1

x(I0×R3) + ‖ulo‖L∞t L6

x(I0×R3) � η3.(11.2)


while from (11.1), (1.16) and (4.5) we see that uhi has small mass:

‖uhi‖L∞t L2

x(I0×R3) � η3.(11.3)

We wish to prove (4.19), or in other words

‖uhi‖L4t L4

x(I0×R3) � η1/41 .(11.4)

By a standard continuity argument,29 it will suffice to show this under thebootstrap hypothesis

‖uhi‖L4t L4

x(I0×R3) ≤ (C0η1)1/4(11.5)

where C0 1 is a large constant (depending only on the energy, and not onany of the η’s). In practice we will overcome this loss of C0 with a positivepower of η3 or η1.

We now use Proposition 10.3 to obtain a Morawetz estimate for φ := uhi.

Theorem 11.1 (Spatially and frequency localized interaction Morawetzinequality). Let the notation and assumptions be as above. Then for anyR ≥ 1,

∫I0

∫R3

|uhi|4 dxdt +∫

I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2|uhi(t, x)|6|x − y| dxdydt � XR,

(11.6)

where XR denotes the quantity

XR := η33

(11.7)

+∫

I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2|ulo(t, x)|5|uhi(t, x)||x − y| dxdydt

(11.8)

+4∑

j=0

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)||PhiO(ujhiu

5−jlo )(t, y)|

(11.9)

· |uhi(t, x)||∇uhi(t, x)| dxdydt

+∫

I0

∫ ∫|x−y|≤2R

|uhi(t, y)||PloO(u5hi)(t, y)||uhi(t, x)||∇uhi(t, x)| dxdydt

(11.10)

29Strictly speaking, one needs to prove that (11.5) implies (11.4) whenever I0 is replacedby any subinterval I1 of I0, in order to run the continuity argument correctly, but it will beclear that the argument below works not only for I0 but also for all subintervals of I0.


+ η1/103

1R

∫I0

( supx∈R3

∫B(x,2R)

|uhi(t, y)|2 dy)dt(11.11)

+1R

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2(|∇uhi(t, x)|2 + |uhi(t, x)|6) dxdydt.(11.12)

Remark 11.2. This should be compared with (1.8). The terms (11.8)–(11.12) may look fearsome, but most of these terms are manageable, becauseof the spatial localization |x − y| ≤ 2R, and because there are not too manyderivatives on the high-frequency term uhi; the only truly difficult terms willbe the last two (11.11), (11.12). Observe from (4.3), (4.4) that we could control(11.12) by (11.11) if we dropped the η

1/103 factor from (11.11); however this

type of factor is indispensable in closing our bootstrap argument, and so wemust treat (11.12) separately. The idea is to use the reverse Sobolev inequality,Proposition 4.8, to control (11.12) by the second term in (11.6), plus an errorof the form (11.11). This can be done, but requires us to apply Theorem 11.1not just for R = 1 (which would be the most natural choice of R) but ratherfor a range of R and then average over such R; see the discussion after theproof of this theorem.

Proof. We apply Phi to (1.1) to obtain

(i∂t + Δ)uhi = Phi(|u|4u),

and then apply Proposition 10.3 with φ := uhi and F := Phi(|u|4u), to obtain

c1

∫I0

∫R3

|uhi(t, x)|4 dxdt

+ c2

∫I0

∫R3

∫R3

|uhi(t, y)|2χ(x − y

R)

x − y

|x − y| · {Phi(|u|4u), uhi}p(t, x) dxdydt

� ‖uhi‖3L∞

t L2x(I0×R3)‖uhi‖L∞

t H1x(I0×R3)

+∫

I0

∫ ∫|x−y|≤2R

|{Phi(|u|4u), uhi}m(t, y)||uhi(t, x)||∇uhi(t, x)| dxdydt

+1R

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2( 1R2

|uhi(t, x)|2 + |∇uhi(t, x)|2) dxdydt.

We now estimate the right-hand side by XR. Observe from (4.3), (11.3)that

‖uhi‖3L∞


t H1x(I0×R3) � η3

3 = (11.7) ≤ XR,

while from (11.3) again we have∫R3

1R2

|uhi(t, x)|2 dx � η23/R2 ≤ η3


and hence1R

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2 1R2

|uhi(t, x)|2 dxdydt � (11.11) ≤ XR.

Similarly we have

1R

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2|∇uhi(t, x)|2 dxdydt � (11.12) ≤ XR.

Now we deal with the mass bracket term. We take advantage of thecancellation (2.6) to write

{Phi(|u|4u), uhi}m = {Phi(|u|4u) − |uhi|4uhi, uhi}m.

We can write, using the notation (1.15),

Phi(|u|4u) − |uhi|4uhi =Phi(|u|4u − |uhi|4uhi) − Plo(|uhi|4uhi)

=4∑

j=0

PhiO(ujhiu

5−jlo ) + PloO(u5

hi).

Thus these terms can be bounded by O((11.9) + (11.10)) = O(XR) (where wetake absolute values everywhere). To summarize so far, we have shown that

c1

∫I0

∫R3

|uhi(t, x)|4 dxdt + c2

∫I0

∫R3

∫R3

|uhi(t, y)|2χ(x − y

R)

x − y

|x − y|· {Phi(|u|4u), uhi}p(t, x) dxdydt � XR.

We now deal with the momentum bracket term, which is a little more in-volved as it requires a little more integration by parts. We will need to exploitthe positivity of one of the components of this term. In order to exploit thecancellation (2.8), we break up the momentum bracket into three pieces:

{Phi(|u|4u), uhi}p

= {|u|4u, uhi}p − {Plo(|u|4u), uhi}p

= {|u|4u, u}p − {|u|4u, ulo}p − {Plo(|u|4u), uhi}p

= {|u|4u, u}p − {|ulo|4ulo, ulo}p − {|u|4u − |ulo|4ulo, ulo}p − {Plo(|u|4u), uhi}p

= −23∇(|u|6 − |ulo|6) − {|u|4u − |ulo|4ulo, ulo}p − {Plo(|u|4u), uhi}p.

We first deal with {Plo(|u|4u), uhi}p estimating the contribution of this termcrudely in absolute values as

O

(∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2|{uhi, Plo(|u|4u)}p(t, x)| dxdydt

).

We wish to bound this term by O((11.11)) = O(XR), recalling that any positivepower of η3 overwhelms any loss in R powers, this follows from


Lemma 11.3.

∫R3 |{uhi, Plo(|u|4u)}p| dx � η

1/23 .

Proof. By (4.3), Holder and Bernstein (1.19),∫R3

|{uhi, Plo(|u|4u)}p| dx �∫

R3

|∇uhi||Plo(|u|4u)| dx+∫

R3

|uhi||∇Plo(|u|4u)| dx

� ‖∇uhi‖L2x‖Plo(|u|4u)‖L2

x+‖uhi‖L2

x‖∇Plo(|u|4u)‖L2

x

� ‖Plo(|u|4u)‖L2x.

Now decompose u = uhi + ulo, and use (1.15) to then decompose

Plo(|u|4u) =5∑

j=0

PloO(ujhiu

5−jlo ).

The terms j = 0, 1, 2, 3, 4 can be estimated by using Bernstein (1.19) and thenHolder to estimate

4∑j=0

‖PloO(ujhiu

5−jlo )‖L2

x�

4∑j=0

‖O(ujhiu

5−jlo )‖L

6/5x

�4∑

j=0

‖uhi‖jL6

x‖ulo‖5−j

L6x

� η3,

thanks to (4.4), (11.1). For the j = 5 term, we argue similarly; indeed we have

‖PloO(u5hi)‖L2

x� ‖O(u5

hi)‖L1x

� ‖uhi‖9/2L6

x‖uhi‖1/2

L2x

� η1/23

by (4.3), (11.1).

Now we deal with the second term in the momentum bracket, namely{|u|4u − |ulo|4ulo, ulo}p. We first move the derivative in (2.3) into a morefavorable position, using the identity

{f, g}p = ∇O(fg) + O(f∇g)

and the identity |u|4u − |ulo|4ulo =∑5

j=1 O(ujhiu

5−jlo ) from (1.15) to write

{|u|4u − |ulo|4ulo, ulo}p =5∑

j=1

(∇O(ujhiu

6−jlo ) + O(uj

hiu5−jlo ∇ulo)).

For the second term, we argue crudely again, estimating this contribution inabsolute value as

O( ∫

I0

5∑j=1

∫ ∫|x−y|≤2R

|uhi(t, y)|2|uhi(t, x)|j |ulo(t, x)|5−j |∇ulo(t, x)| dxdydt).

(11.13)

But by (4.4), (11.1) and the low frequency localization of ulo we have∫R3

|uhi|j |ulo|5−j |∇ulo| dx�‖uhi‖jL6

x‖ulo‖5−j

L6x‖∇ulo‖L6

x�‖uhi‖j

L6x‖ulo‖6−j

L6x

�η3,


and so this term can also be bounded by O((11.11)) = O(XR). For the firstterm, we can integrate by parts and then take absolute values to estimate thecontribution of this term by

5∑j=1

O( ∫

I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2|uhi(t, x)|j |ulo(t, x)|6−j

|x − y| dxdydt).(11.14)

Note that the terms where the integration by parts hits the cutoff is of the sametype, but with the 1

|x−y| factor replaced by O( 1R) on the region where |x−y|∼R;

it is clear that this term is dominated by (11.14) (or by a variant of (11.13),where we remove the ∇ from ulo). We hold off on estimating this term fornow, and turn to the first term in the momentum bracket: −2

3∇(|u|6 − |ulo|6).After an integration by parts, this term can be written as

c3

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2(|u(t, x)|6 − |ulo(t, x)|6)|x − y| dxdydt

+ O( 1

R

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2||u(t, x)|6 − |ulo(t, x)|6|| dxdydt),

for some explicit constant c3 > 0; note that the error incurred by removing thecutoff χ(x−y

R ) by the cruder cutoff |x − y| ≤ 2R can be controlled by (11.12)which is acceptable. To control the error term, we use (1.14) to split

|u(t, x)|6 − |ulo(t, x)|6 = |uhi(t, x)|6 +5∑

j=1

O(ujhi(t, x)u6−j

lo (t, x)).(11.15)

The |uhi|6 term is of course bounded by O((11.12)) = O(XR). The remainingterms have an L1

x norm of O(η3) by (4.4), (11.1), and Holder, so that this errorterm is bounded by O((11.11)) = O(XR). For the main term, we again use(11.15) and observe that the contributions of the error terms are bounded by(11.14). Collecting all these estimates together, we have now shown that

c1

∫I0

∫R3

|uhi(t, x)|4 dxdt

+ c3

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2|uhi(t, x)|6|x − y| dxdydt � XR

+5∑

j=1

O(∫

I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2|uhi(t, x)|j |ulo(t, x)|6−j

|x − y| dxdydt).

We can eliminate the j = 2, 3, 4, 5 terms using the elementary inequality

|uhi(t, x)|j |ulo(t, x)|6−j ≤ ε|uhi(t, x)|6 + C(ε)|uhi(t, x)||ulo(t, x)|5

for some small absolute constant ε; this allows us to control the j = 2, 3, 4, 5terms by the j = 1 term, plus a small multiple of the j = 6 term which can thenbe absorbed by the second term on the left-hand side (shrinking c3 slightly).The theorem follows.


We would like to use Theorem 11.1 to prove (11.4). If it were not for theerror terms (11.8)–(11.12) then this estimate would follow immediately fromTheorem 11.1, since we can discard the second term in (11.6) as being positive.One would then hope to estimate all of the error terms (11.8)–(11.12) by O(η1)using the bootstrap hypothesis (11.5) and the other estimates available (e.g.(4.3), (4.4), (11.1), (11.3)). It turns out that this strategy works (with R = 1)for the first four error terms (11.8)–(11.11) but not for the fifth term (11.12).To estimate this term we need the reverse Sobolev inequality, which effectivelyreplaces the |∇uhi|2 density here by |uhi|6, at which point one can hope tocontrol this term by the second positive term in (11.6). But to do so it turnsout that one cannot apply Theorem 11.1 for a single value of R, but mustinstead average over a range of R.

We turn to the details. We let J = J(η1, η2) 1 be a large30 integer, andapply Theorem 11.1 to all dyadic radii R = 1, 2, . . . , 2J separately. We thenaverage over R to obtain∫

I0

∫R3

|uhi|4 dxdt + Y � X,(11.16)

where Y is the positive quantity

Y :=1J

∑1≤R≤2J

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2|uhi(t, x)|6|x − y| dxdydt

(R is always understood to sum over dyadic numbers), and X is the quantity

X := η33

(11.17)

+ sup1≤R≤2J

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2|ulo(t, x)|5|uhi(t, x)||x − y| dxdydt

(11.18)

+ sup1≤R≤2J

4∑j=0

(11.19)

·∫

I0

∫ ∫|x−y|≤2R


5−jlo )(t, y)||uhi(t, x)||∇uhi(t, x)| dxdydt

+ sup1≤R≤2J

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)||PloO(u5hi)(t, y)||uhi(t, x)||∇uhi(t, x)| dxdydt

(11.20)

30But it is not too large; any factor of η3 will easily be able to overcome any losses dependingon J .


+ η1/103 sup

1≤R≤2J

1R

∫I0

(supx∈R3

∫B(x,2R)

|uhi(t, y)|2 dy)dt

(11.21)

+1J

∑1≤R≤2J

1R

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2(|∇uhi(t, x)|2 + |uhi(t, x)|6) dxdydt,

(11.22)

where we have estimated the average 1J

∑1≤R≤2J by the supremum in those

terms for which the averaging is not important.31

The terms (11.17)–(11.22) are roughly arranged in increasing order ofdifficulty to estimate. But let us use the reverse Sobolev inequality already ob-tained in Proposition 4.8 to deal with the most difficult term (11.22), replacingit by easier terms.

Lemma 11.4.

(11.21) + (11.22) � η1/1001 (Y + W )

where

W := sup1≤R≤C(η1,η2)2J

1R

∫I0

( ∑x∈ R

100Z3

( ∫B(x,3R)

|uhi(t, y)|2 dy)100)1/100

dt,

(11.23)

and R100Z3 is the integer lattice Z3 dilated by R/100.

Proof. We first handle (11.21). For every x ∈ R3 there exists x′ ∈ R100Z3

such that B(x, 2R) is contained in B(x′, 3R). Thus

1R

supx∈R3

∫B(x,2R)

|uhi(t, y)|2 dy � 1R

( ∑x′∈ R

100Z3

( ∫B(x′,3R)

|uhi(t, y)|2 dy)100)1/100

,

and the claim follows by noting that η1/103 � η

1/1001 .

Now we consider (11.22), writing this term as

1J

∑1≤R≤2J

1R

∫I0

∫R3

|uhi(t, y)|2ehi(t, y, 2R) dydt,

where ehi(t, y, 2R) is the local energy,

ehi(t, y, 2R) :=∫

B(y,2R)|∇uhi(t, x)|2 + |uhi(t, x)|6 dx.

We will denote the same quantity with uhi replaced with u by e(t, y, 2R). Wesplit this term into the regions ehi(t, y, 2R) � η1 and ehi(t, y, 2R) η1.

31Indeed, in those terms we will extract a gain of η3 which will easily absorb any lossesrelating to R = O(2J), since J depends only on η1 and η2.


Large energy regions. Consider first the large energy regions ehi(t, y, 2R) η1. By (11.2), the same lower bound holds for e(t, y, 2R). For (t, y) in suchregions, we apply Proposition 4.8 to conclude that∫

B(y,2R)|∇u(t, x)|2dx ≤ 1

2e(t, y, 2R) + C(η1, η2)

∫B(y,C(η1,η2)R)

|u(t, x)|6dx

which implies∫B(y,2R)

|∇u(t, x)|2dx � C(η1, η2)∫

B(y,C(η1,η2)R)|u(t, x)|6dx.

The same estimate is valid for uhi in light of (11.2). Thus we can bound thecontribution to (11.22) of the large energy regions by

� C(η1, η2)1J

∑1≤R≤2J

1R

∫I0

∫ ∫|x−y|�C(η1,η2)R

|uhi(t, y)|2|uhi(t, x)|6 dxdydt.

Shifting R by C(η1, η2), we can bound this by

� (C(η1, η2))21J

∑1≤R≤C(η1,η2)2J

AR(11.24)

whereAR :=

1R

∫I0

∫ ∫|x−y|≤R

|uhi(t, y)|2|uhi(t, x)|6 dxdydt.

To bound this by η1/1001 (Y + W ) (and not just by O(Y )) we exploit the aver-

aging32 in R. First observe that∑1≤R≤R′

AR �∫

I0

∫ ∫|x−y|≤R′

|uhi(t, y)|2|uhi(t, x)|6|x − y| dxdydt

for all R′. Averaging this over all 1 ≤ R′ ≤ 2J , we see that

1J

∑1≤R′≤2J

∑1≤R≤R′

AR

� 1J

∑1≤R′≤2J

∫I0

∫ ∫|x−y|≤R′

|uhi(t, y)|2|uhi(t, x)|6|x − y| dxdydt � Y.

Now let 1 < J0 < J be a parameter depending on η1, η2 to be chosen shortly.Observe that for each 1 ≤ R ≤ 2J−J0 there are at least J0 values of R′ whichinvolve that value of R in the above sum. Thus we have

J0

J

∑1≤R≤2J−J0

AR � Y,

32The key point here is that while the AR quantities have a factor of 1/R, the quantitiesin Y have a larger factor of 1/|x − y|. After averaging, the latter factor begins to dominatethe former.


and thus the contribution to (11.24) of the terms where R ≤ 2J−J0 is boundedby O( (C(η1,η2))2

J0Y ), which is acceptable if J0 is chosen sufficiently large depend-

ing on η1, η2.It remains to control the expression

(C(η1, η2))2

J

∑2J−J0≤R≤C(η1,η2)2J

1R

∫I0

∫ ∫|x−y|≤R

|uhi(t, y)|2|uhi(t, x)|6dxdydt.

This is bounded by

C(η1, η2, J0)J

sup1≤R≤C(η1,η2)2J

1R

∫I0

∫supx∈R3

( ∫B(x,R)

|uhi(t, y)|2dy)|uhi(t, x)|6dxdt.

Using the energy bound (4.4), we see that this is in turn bounded by

� C(η1, η2, J0)J

sup1≤R≤C(η1,η2)2J

1R

(∫I0

supx∈R3

( ∫B(x,R)

|uhi(t, y)|2dy)dt

)� 1

JC(η1, η2, J0)W

� η1/1001 W

when J = J(η1, η2) is sufficiently large.

Small energy regions. Now we deal with the contribution of the lowenergy regions:

1J

∑1≤R≤2J

1R

∫I0

∫ehi(t,y,2R)�η1

|uhi(t, y)|2ehi(t, y, 2R) dydt.

Observe that if y is such that ehi(t, y, 2R) � η1, then there exists y′ ∈ R100Z3

such that y ∈ B(y′, R) ⊆ B(y, 2R) ⊂ B(y′, 3R), and thus ehi(t, y, R) �min(η1, ehi(t, y′, 3R)). Thus we can dominate the above expression by

� 1J

∑1≤R≤2J

1R

∫I0

∑y′∈ R

100Z3

min(η1, ehi(t, y′, 3R))∫

B(y′,3R)|uhi(t, y)|2 dydt.

Now from (4.3) we have, using min(η1, ehi(·))100/99 ≤ η1/991 ehi(·), that⎛⎝ ∑

y′∈ R

100Z3

min(η1, ehi(t, y′, 3R))100/99

⎞⎠99/100

� η1/1001

⎛⎝ ∑y′∈ R

100Z3

ehi(t, y′, 3R)

⎞⎠99/100

� η1/1001 .

Thus, by Holder’s inequality we can bound the previous expression byO(η1/100

1 W ) as desired.


In light of all the above estimates, we have thus shown that∫I0

∫R3

|uhi|4 dxdt � η33 + (11.18) + (11.19) + (11.20) + η

1/1001 W,(11.25)

since the Y term on the left can be used to absorb the η1/1001 Y term which would

otherwise appear on the right. It thus suffices to show that all of the quantities(11.18)-(11.20) are O(η1), while the factor of η

1/1001 allows the quantity W to

be estimated using the weaker bound of O(C0η1). (This is the main purpose ofthe reverse Sobolev inequality, Proposition 4.8, in our argument. The constantC0 was defined in our bootstrap assumption (11.5).).

As mentioned earlier, the terms (11.18)–(11.20) are roughly arranged inincreasing order of difficulty, and W is more difficult still. To estimate anyof these expressions, we of course need good spacetime estimates on uhi andulo. We do already have some estimates on these quantities (11.5), (4.3),(4.4), (11.2), (11.3), but it turns out that these are not directly sufficient toestimate (11.18)–(11.20) and W . Thus we shall first use the equation (1.1) andStrichartz estimates to bootstrap (11.5) to yield further spacetime integrability;this will be the purpose of the next section.

12. Interaction Morawetz: Strichartz control

In this section we establish the spacetime estimates we need in order tobound (11.18)–(11.20) and W . Ideally one would like to use (11.5) to showthat u obeys the same estimates as a solution to the free Schrodinger equation;however the quantity (11.5) is supercritical (it has roughly the scaling of H1/4,instead of H1), and we must therefore accept some loss of derivatives in thehigh frequencies;33 a model example of a function u obeying (11.5) to keepin mind is a pseudosoliton solution where u has magnitude |u(t, x)| ∼ N1/2

on the spacetime region x = O(N−1), t = O(N) and has Fourier transformsupported near the frequency N for some large N 1. We will however beable to show that u does behave like a solution to the free Schrodinger equationmodulo a high frequency forcing term which is controlled in L2

t L1x but not in

dual Strichartz spaces.Recall that the constant C0 which we use throughout these next three

sections of the paper was defined in (11.5), our bootstrap hypothesis on theL4 norm of uhi. We begin by estimating the low frequency portion ulo of thesolution, which does behave like the free equation from the point of view ofStrichartz estimates:

33Recall that uhi and ulo were defined at the beginning of Section 11.


Proposition 12.1 (Low frequency estimate). For the functions uhi, ulo

defined at the start of Section 11,

‖∇ulo‖L2t L6

x(I0×R3) � C1/20 η

1/21(12.1)

and

‖ulo‖L4t L∞

x (I0×R3) � η1/23 .(12.2)

In fact, there is the slightly more general statement that (12.1), (12.2) hold forall u≤N when N ∼ 1.

Observe that the L4t L

∞x estimate gains a power of η3, which will be very

useful for us in overcoming certain losses of R in the sequel.

Proof. We may assume that N ≥ 1, since the case when N < 1 of coursethen follows. Let Z denote the quantity

Z := ‖∇u≤N‖L2t L6

x(I0×R3) + ‖u≤N‖L4tL∞

x (I0×R3).(12.3)

By Strichartz (3.7), (3.4) we have

Z � ‖u≤N‖S1(I0×R3) � ‖u≤N (t0)‖H1x

+ ‖∇P≤N (|u|4u)‖L2tL

6/5x (I0×R3).

By (11.1) we have‖u≤N (t0)‖H1

x� η3.

Now we consider the nonlinear term. We split u = u≤N + u>N and use (1.15)to write

∇P≤N (|u|4u) =5∑

j=0

∇P≤NO(uj>Nu5−j

≤N ).

Considering the j = 0 term first, we discard P≤N and use the Leibnitz ruleand Holder to estimate

‖∇P≤NO(u5≤N )‖L2

t L6/5x (I0×R3) � ‖O(u4

≤N∇u≤N )‖L2t L

6/5x (I0×R3)

� ‖u≤N‖4L∞

t L6x(I0×R3)‖∇u≤N‖L2

tL6x(I0×R3);

by (11.1) and (12.3) this term is thus bounded by O(η43Z).

Now consider the j = 1 term. We argue similarly to estimate this term as

‖∇P≤NO(u4≤Nu>N )‖L2

tL6/5x (I0×R3)

� ‖O(u4≤N∇u>N )‖L2

tL6/5x (I0×R3)

+ ‖O(u3≤Nu>N∇u≤N )‖L2

t L6/5x (I0×R3)

� ‖u≤N‖2L4

t L∞x (I0×R3)‖u≤N‖2

L∞t L6

x(I0×R3)‖∇u>N‖L∞t L2

x(I0×R3)

+ ‖u≤N‖3L∞

t L6x(I0×R3)‖u>N‖L∞

t L6x(I0×R3)‖∇u≤N‖L2

t L6x(I0×R3);

by (4.4), (11.1), (12.3) these terms are thus bounded by O(η23Z

2 + η33Z).


Now consider the j = 2, 3, 4, 5 terms. This time we use Bernstein’s in-equality (1.20) (recalling that N ∼ 1) and Holder to estimate

‖∇P≤NO(u5−j≤N uj

>N )‖L2t L

6/5x (I0×R3)

� ‖O(u5−j≤N uj

>N )‖L2t L1

x(I0×R3)

� ‖u>N‖2L4

t L4x(I0×R3)‖u>N‖j−2

L∞t L6

x(I0×R3)‖u≤N‖5−jL∞

t L6x(I0×R3).

Applying (11.5), (4.4) we can bound these terms by O(C1/20 η

1/21 ) (we can do

significantly better on the j = 2, 3, 4 terms but we will not exploit this). Com-bining all these estimates we see that

Z � η3 + η43Z + η2

3Z2 + η3

3Z + C1/20 η

1/21 ;

by standard continuity arguments this then implies that Z � C1/20 η

1/21 . This

proves (12.1), but we did not achieve the η3 gain in (12.2).To obtain (12.2) we must refine the above analysis. Let u0

≤N be thesolution to the free Schrodinger equation with initial data u0

≤N (t0) = u≤N (t0)for some t0 ∈ I0. Then by (11.1) and Lemma 3.1

‖u0≤N‖L4

t L∞x (I0×R3) � η3.

Thus it suffices to prove that

‖u≤N − u0≤N‖L4

t L∞x (I0×R3) � η

1/23 .

We estimate the left-hand side as∑N ′�N

‖uN ′ − u0N ′‖L4

t L∞x (I0×R3).

By Bernstein (1.20) we may bound this by∑N ′�N

N ′‖uN ′ − u0N ′‖L4

t L3x(I0×R3)

which by interpolation can be bounded by∑N ′�N

N ′‖uN ′ − u0N ′‖1/2

L2tL6

x(I0×R3)‖uN ′ − u0N ′‖1/2

L∞t L2

x(I0×R3).

But from (11.1),

‖uN ′ − u0N ′‖L∞

t L2x(I0×R3) � (N ′)−1‖u≤N − u0

≤N‖L∞t H1(I0×R3) � (N ′)−1η3

so that

‖u≤N − u0≤N‖L4

t L∞x (I0×R3) � η

1/23

∑N ′≤N

(N ′)1/2‖uN ′ − u0N ′‖1/2

L2t L6

x(I0×R3).


Applying Strichartz (3.7) yields

‖u≤N − u0≤N‖L4

t L∞x (I0×R3) � η

1/23

∑N ′≤N

(N ′)1/2‖PN ′(|u|4u)‖1/2

L2t L

6/5x (I0×R3)

.

But the preceding analysis already showed that

‖PN ′(|u|4u)‖L2t L

6/5x (I0×R3) � η4

3Z + η23Z

2 + η33Z + C

1/20 η

1/21 � 1

(for instance), and the claim follows.

Now we estimate the high frequencies. We first need an estimate on thehigh-frequency portion of the nonlinearity |u|4u. It turns out that this quan-tity cannot be easily estimated in a single Strichartz norm, but must insteadbe decomposed into two pieces estimated using separate space-time Lebesguenorms (cf. the appearance of M in Lemma 3.2).

Proposition 12.2. We can decompose

Phi(|u|4u) = F + G

where F , G are Schwartz functions with Fourier support in the region |ξ| � 1and

‖∇F‖L2tL

6/5x (I0×R3) � η

1/23

and‖G‖L2

t L1x(I0×R3) � C

1/20 η

1/21 .

Of the two pieces, F is by far the better term; indeed, if G were not present,then the Strichartz estimate (3.7) would be able to obtain L10

t,x bounds on uhi.The reader may in fact assume as a first approximation that F is negligible,and that the nonlinearity Phi(|u|4u) is primarily in L2

t L1x, which is not a dual

S1 Lebesgue norm. Note also that the η121 bound ultimately determines the η1

on the right side of (4.19) at the end of Section 14.

Proof. We split u = ulo + uhi, and then use (1.15) to split

Phi(|u|4u) =5∑

j=0

PhiO(ujhiu

5−jlo ).

Considering the j = 0 term first, we have

‖∇PhiO(u5lo)‖L2

t L6/5x (I0×R3)

� ‖O(u4lo∇ulo)‖L2

t L6/5x (I0×R3)

� ‖ulo‖2L4

tL∞x (I0×R3)‖ulo‖2

L∞t L6

x(I0×R3)‖∇ulo‖L∞t L2

x(I0×R3)

which is O(η1/23 ) (for instance) by Proposition 12.1 and (4.4), (4.3). Thus this

term may be placed as part of F .


Now consider the j = 1 term. We have

‖∇PhiO(u4louhi)‖L2

t L6/5x (I0×R3)

� ‖O(u4lo∇uhi)‖L2

tL6/5x (I0×R3)

+ ‖O(u3louhi∇ulo)‖L2

t L6/5x (I0×R3)

� ‖ulo‖2L4

t L∞x (I0×R3)‖ulo‖2

L∞t L6

x(I0×R3)‖∇uhi‖L∞t L2

x(I0×R3)

+ ‖ulo‖3L∞


t L6x(I0×R3)‖∇ulo‖L2

tL6x(I0×R3).

Applying Proposition 12.1, (4.4), (4.3), and (11.1) this expression is O(η33) and

so this term may also be placed as part of F .Now consider the j = 2, 3, 4, 5 terms. We estimate

‖PhiO(ujhiu

5−jlo )‖L2

t L1x(I0×R3)

� ‖uhi‖2L4

t L4x(I0×R3)‖uhi‖j−2

L∞t L6

x(I0×R3)‖ulo‖5−jL∞

t L6x(I0×R3)

which is O(C1/20 η

1/21 ) by (11.5) and (4.4). Thus we may place this term as part

of G.

Corollary 12.3. For every N ≥ 1,

‖uN‖L2t L6

x(I0×R3) � C1/20 N1/2η

1/21 .(12.4)

Proof. From Strichartz (3.7) and Proposition 12.2,

‖uN‖L2t L6

x(I0×R3) � ‖uN (t0)‖L2x(R3) + ‖PNF‖L2

t L6/5x (I0×R3) + ‖PNG‖L2

t L6/5x (I0×R3).

The first term is certainly acceptable by (11.3). The second term is O(η1/23 N−1)

by Proposition 12.2, and the third term is O(C1/20 N1/2η

1/21 ) by Bernstein (1.20)

and Proposition 12.2. The claim follows.

13. Interaction Morawetz: Error estimates

We now show that the comparatively easy terms (11.18), (11.19), (11.20)are indeed controlled by O(η1), which is in turn bounded by the right-handside of (11.4). The term W is however significantly harder and will be deferredto the next section.

• The estimation of (11.18). We have to show that∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)|2|ulo(t, x)|5|uhi(t, x)||x − y| dxdydt � η1

for all 1 ≤ R ≤ 2J . This term will be fairly easy because of the localization|x − y| ≤ 2R. Observe that the kernel 1

|x| has an L1x norm of O(R2) ≤ O(22J)


on the ball B(0, 2R). Thus by Young’s inequality and Cauchy-Schwarz we seethat ∫ ∫

|x−y|≤2R

F (x)G(y)|x − y| dxdy � R2‖F‖L2

x‖G‖L2

x

for any functions F , G. In particular the expression to be estimated is boundedby

� 22J

∫I0

‖uhi(t)‖2L4

x‖|ulo|5|uhi|(t)‖L2

xdt.

We use Holder and (11.3) to estimate

‖|ulo|5|uhi|(t)‖L2x

� ‖ulo(t)‖5L∞

x‖uhi(t)‖L2

x� η3‖ulo(t)‖5

L∞x

.

We dispose of three of the five factors of ‖ulo(t)‖5L∞

xby observing from (4.4) and

Bernstein’s inequality (1.20) that ‖ulo(t)‖3L∞

x� η3

3 � 1. Combining all theseestimates and then using Holder in time, we thus can bound the expression tobe estimated by

22Jη3‖uhi‖2L4

tL4x(I0×R3)‖ulo‖2

L4t L∞

x (I0×R3),

which is bounded by the right-hand side of (11.4) using (11.5) and Proposition12.1 (note that η3 will absorb 22J since J depends only on η1 and η2). Thisconcludes the treatment of (11.18).

• The estimation of (11.19). We now handle (11.19). We have to showthat∫

I0

∫ ∫|x−y|≤2R


5−jlo )(t, y)||uhi(t, x)||∇uhi(t, x)| dxdydt � η1

for j = 0, 1, 2, 3, 4 and 1 ≤ R ≤ 2J .We begin by considering the cases j = 1, 2, 3, 4. We observe from Holder

and (4.3) that∫B(y,2R)

|uhi(t, x)||∇uhi(t, x)| dx �R3/4‖uhi(t)‖L4x‖∇uhi(t)‖L2

x

�R3/4‖uhi(t)‖L4x,

and hence it suffices to show that

R34

∫I0

‖uhi(t)‖L4x

∫R3


5−jlo )(t, y)| dydt � η1.

We use Holder (and the hypothesis j = 1, 2, 3, 4) to estimate this as

� R34 ‖uhi‖3

L4t L4

x(I0×R3)‖ulo‖L4tL∞

x (I0×R3)‖uhi‖j−1L∞

t L6x(I0×R3)‖ulo‖4−j

L∞t L6

x(I0×R3).

The L∞t L6

x(I0 × R3) factors are bounded by (4.4), leaving us to prove

R34 ‖uhi‖3

L4tL4

x(I0×R3)‖ulo‖L4t L∞

x (I0×R3) � η1.


But this follows from (11.5) and Proposition 12.1, again using η3 to wallop apositive power of R.

Finally, we consider the j = 0 case of (11.19), where we have to prove∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)||PhiO(u5lo)(t, y)||uhi(t, x)||∇uhi(t, x)| dxdydt � η1.

Here we use Cauchy-Schwarz and (11.3) to crudely bound∫R3

|uhi(t, x)||∇uhi(t, x)| dx � η3

and then use Holder to reduce to showing that

η3‖uhi‖L∞t L2

x(I0×R3)‖PhiO(u5lo)‖L1

t L2x(I0×R3) � η1.

The factor ‖uhi‖L∞t L2

x(I0×R3) is O(η3) by (11.3). For the second factor, we takeadvantage of the high frequency localization (using (1.16)) to write

‖PhiO(u5lo)‖L1

tL2x(I0×R3) � ‖∇O(u5

lo)‖L1t L2

x(I0×R3)

� ‖O(u4lo∇ulo)‖L1

t L2x(I0×R3)

� ‖∇ulo‖L∞t L2

x(I0×R3)‖ulo‖4L4

t L∞x (I0×R3).

The claim then follows from (4.3) and Proposition 12.1.

• The estimation of (11.20). We now handle (11.20). We have to showthat

∫I0

∫ ∫|x−y|≤2R

|uhi(t, y)||PloO(u5hi)(t, y)||uhi(t, x)||∇uhi(t, x)| dxdydt � η1.

(13.1)

We begin by using Holder to write∫B(y,2R)

|uhi(t, x)||∇uhi(t, x)| dx � R1/2‖uhi(t)‖L3x‖∇uhi(t)‖L2

x

and apply (4.3) to estimate the left-hand side of (13.1) by

� R12

∫I0

‖uhi(t)‖L3x

∫|uhi(t, y)||PloO(u5

hi)(t, y)| dy.

By Holder and Bernstein (1.20) we have∫|uhi(t, y)||PloO(u5

hi)(t, y)| dy � ‖uhi(t)‖L3x‖PloO(uhi(t)5)‖L

3/2x

� ‖uhi(t)‖L3x‖O(uhi(t)5)‖L1

x

� ‖uhi(t)‖L3x‖uhi(t)‖2

L4x‖uhi(t)‖3

L6x.


The terms ‖uhi(t)‖3L6

xare bounded by (4.4), so by a Holder in time and (11.5)

we have

(11.20)� R12 ‖uhi‖2

L4tL3

x(I0×R3)‖uhi‖2L4

t L4x(I0×R3)(13.2)

� R1/2C1/20 η

1/21 ‖uhi‖2

L4t L3

x(I0×R3).

From the triangle inequality and Holder,

‖uhi‖L4t L3

x(I0×R3) �∑N≥1

‖uN‖L4t L3

x(I0×R3)

�∑N≥1

‖uN‖1/2L∞

t L2x(I0×R3)‖uN‖1/2

L2t L6

x(I0×R3).

From (11.3), (4.5) we have ‖uN‖L∞t L2

x(I0×R3) � min(η3, N−1). Applying (12.4)

we thus have

‖uhi‖L4t L3

x(I0×R3) �∑N≥1

min(η3, N−1)1/2C

1/40 N1/4η

1/41 � η

1/23 C

1/40 η

1/41 .

Inserting this estimate into (13.2) we see that (11.20) is acceptable (again, thepower of η3 counteracts the loss in C0 and the presence of the power R1/2).

Note that the first four factors on the right side of (11.25) have all in factbeen shown to be controlled with a positive power of η3.

14. Interaction Morawetz: A double Duhamel trick

To conclude the proof of Proposition 4.9, we have to show that

W � C0η1,

or in other words that1R

∫I0

( ∑x∈ R

100Z3

( ∫B(x,3R)

|uhi(t, y)|2 dy)100)1/100

dt � C0η1

for all 1 ≤ R ≤ C(η1, η2)2J . By duality we have( ∑x∈ R

100Z3

( ∫B(x,3R)

|uhi(t, y)|2 dy)100)1/100

=∑

x∈ R

100Z3

c(t, x)∫

B(x,3R)|uhi(t, y)|2 dy

where c(t, x) > 0 are a collection of numbers which are almost summable inthe sense that ∑

x∈ R

100Z3

c(t, x)100/99 = 1(14.1)

for all t. Thus it suffices to show that1R

∫I0

∑x∈ R

100Z3

c(t, x)∫

B(x,3R)|uhi(t, y)|2 dydt � C0η1.


Let ψ be a bump function adapted to B(0, 5) which equals 1 on B(0, 3). Since

∫B(x,3R)

|uhi(t, y)|2 dy �∫

|uhi(t, y)|2ψ(

y − x

R

)dy,

it suffices to show that

1R

∫I0

∑x∈ R

100Z3

c(t, x)∫

R3

|uhi(t, y)|2ψ(

y − x

R

)dydt � C0η1.(14.2)

In the proof of Lemma 11.4, we obtained spacetime control on uhi by usingthe (forward-in-time) Duhamel formula (1.13) followed by Strichartz estimates.This seems to be insufficient to prove (14.2) (the best argument available seemsto lose a logarithm of the derivative); instead, we rely on both the forward-in-time and backward-in-time Duhamel formulae (1.13), and argue using thefundamental solution (1.11). This will only lose a factor of C0 (see (11.5)),which is acceptable because of the gain of η

1/1001 which was obtained earlier in

Lemma 11.4.Let us write I0 = [t−, t+] for some times −∞ < t− < t+ < ∞. We use

Proposition 12.2 to decompose Phi(|u|4u) = F + G. We define the functionsu±

hi to solve the Cauchy problem

(i∂t + Δ)u±hi = F ; u±

hi(t±) = uhi(t±);

thus this is the same equation as uhi satisfies but without the term G. From(1.13) observe the forward-in-time Duhamel formula

uhi(t) = u−hi(t) − i

∫t−<s−<t

ei(t−s−)ΔG(s−) ds−

and the backward-in-time Duhamel formula

uhi(t) = u+hi(t) + i

∫t<s+<t+

ei(t−s+)ΔG(s+) ds+.

Let us see how we would prove (14.2) if uhi were replaced by u±hi. From

Strichartz (3.7), (11.3), and Proposition 12.2 (discarding a derivative) we seethat

‖u±hi‖L2

t L6x(I0×R3) � η

1/23 .


But from Holder we have1R

∑x∈ R

100Z3

c(t, x)∫

R3

|u±hi(t, y)|2ψ

(y − x

R

)dy

� 1R

∑x∈ R

100Z3

c(t, x)R2

(∫R3

|u±hi(t, y)|6ψ

(y − x

R

)dy

)1/3

� R

⎛⎝ ∑x∈ R

100Z3

c(t, x)3/2

⎞⎠2/3 ⎛⎝ ∑x∈ R

100Z3

∫R3

|u±hi(t, y)|6ψ

(y − x

R

)dy

⎞⎠1/3

� R

⎛⎝ ∑x∈ R

100Z3

c(t, x)100/99

⎞⎠99/100

‖u±hi‖2

L6x

= R‖u±hi‖2

L6x

and hence1R

∫I0

∑x∈ R

100Z3

c(t, x)∫

R3

|uhi(t, y)|2ψ(

y − x

R

)dydt � Rη3

which is acceptable if η3 is sufficiently small (recall that R ≤ C(η1, η2)2J andJ depends only on η1, η2). Thus we see that (14.2) would be easy to prove ifuhi were replaced by u±

hi.It is now natural to use one of the two Duhamel formulae listed above,

and attempt to prove (14.2) for the integral term. This however turns outto be rather difficult. It will be significantly easier if we use both formulaesimultaneously. More precisely, we re-arrange the above Duhamel formulae as

−i

∫t−<s−<t

ei(t−s−)ΔG(s−) ds− = uhi(t) − u−hi(t)

andi

∫t<s+<t+

ei(t−s+)ΔG(s+) ds+ = uhi(t) − u+hi(t).

Then we multiply the first identity by the conjugate of the second to obtain

−∫ ∫

t−<s−<t<s+<t+

(ei(t−s−)ΔG(s−))(ei(t−s+)ΔG(s+)) ds+ ds−

= |uhi(t)|2 − u−hi(t)uhi(t) − uhi(t)u+

hi(t) + u−hi(t)u

+hi(t).

From the elementary pointwise estimates

|u−hi(t)uhi(t)| ≤

14|uhi(t)|2 + O(|u−

hi(t)|2),

|uhi(t)u+hi(t)| ≤

14|uhi(t)|2 + O(|u+

hi(t)|2),

|u−hi(t)u

+hi(t)| ≤ O(|u−

hi(t)|2) + O(|u+hi(t)|2)


we thus have the pointwise inequality

|uhi(t)|2 �∣∣∣ ∫ ∫

t−<s−<t<s+<t+

ei(t−s−)ΔG(s−)ei(t−s+)ΔG(s+) ds−ds+

∣∣∣(14.3)

+|u−hi(t)|2 + |u+

hi(t)|2.

This should be compared with what one would obtain with a single Duhamelformula (1.13), namely something like

|uhi(t)|2 �∣∣∣ ∫ ∫

t−<s,s′<tei(t−s)ΔG(s)ei(t−s′)ΔG(s′) dsds′

∣∣∣ + |u−hi(t)|2.

This turns out to be an inferior formulation; the basic problem is that theintegral

∫t−<s,s′<t

dsds′

|s−s′| is logarithmically divergent, whereas the integral∫t−<s−<t<s+<t+

ds−ds+

|s−−s+| is not.We now return to (14.2), and insert (14.3). The latter two terms were

already shown to be acceptable. So we are left to prove that

1R

∣∣∣ ∫ ∫ ∫t−<s−<t<s+<t+

∑x∈ R

100Z3

c(t, x)∫

R3

ei(t−s−)ΔG(s−)(y)ei(t−s+)ΔG(s+)(y)

· ψ(y − x

R) dyds−ds+dt

∣∣∣ � C0η1.

(14.4)

To compute the y integral, we use the following stationary phase estimate.

Lemma 14.1. For any t− < s− < t < s+ < t+, and any Schwartz func-tions f−(x), f+(x),∣∣∣ ∑

x∈ R

100Z3

c(t, x)∫

R3

ei(t−s−)Δf−(y)ei(t−s+)Δf+(y)ψ(y − x

R) dy

∣∣∣� |s+ − s−|−3/2 min(θ−3/2+3/100, 1)‖f−‖L1

x‖f+‖L1

x

where θ := |t−s+||t−s−|R2|s+−s−| .

Proof. Fix t. We use the explicit formula for the fundamental solution(1.11) to estimate the left-hand side as

� 1|t − s+|3/2|t − s−|3/2

∑x∈ R

100Z3

c(t, x)∫

R3

∫R3(∫

R3

ei(|y−x−|2/(t−s−)−|y−x+|2/(t−s+))ψ

(y − x

R

)dy

)f+(x+)f−(x−) dx+dx−,


so it suffices to show that∣∣∣ ∑x∈ R

100Z3

c(t, x)∫

R3

ei(|y−x−|2/(t−s−)−|y−x+|2/(t−s+))ψ(y − x

R)dy

∣∣∣� |t − s+|3/2|t − s−|3/2

|s+ − s−|3/2min(θ−3/2+3/100, 1)

= R3 min(θ3/100, θ3/2)

for all x−, x+. With the change of variables y = x + Rz, this becomes∣∣∣ ∑x∈ R

100Z3

c(t, x)I(x)∣∣∣ � min(θ3/100, θ3/2),

where the integrals I(x) are defined by

I(x) :=∫

R3

eiΦx(z)ψ(z)dz

and Φx = Φx,R,x−,x+,s−,s+,t is the phase

Φx(z) := |x − x− + Rz|2/(t − s−) − |x − x+ + Rz|2/(t − s+).

By the normalization of c(t, x), it thus suffices to prove the bounds( ∑x∈ R

100Z3

I(x)100)1/100

� min(θ3/100, θ3/2).

We now divide into two cases, depending on the size of θ. First suppose thatθ 1. Observe that the gradient of the phase Φx in z is

∇zΦx(z) = 2R(x − x− + Rz)/(t − s−) − 2R(x − x+ + Rz)/(t − s+)

= 2R(x + Rz − x∗)s− − s+

(t − s−)(t − s+)

where x∗ = x∗(s−, s+, t, x−, x+, R) is a quantity not depending on x or z. Inparticular, in the region where

|x − x∗| |t − s−||t − s+|

R|s+ − s−|= Rθ R

we can obtain an extremely good bound from the principle of nonstationaryphase,34 namely that

|I(x)| �( |x − x∗|

Rθ

)−100

.(14.5)

34In other words, one can use repeated integration by parts; see [41]. An alternate approach

in this lemma is to use a Gaussian cutoff e−π|x|2 instead of ψ, and then compute all theintegrals explicitly by contour integration (or equivalently by using the “Gaussian beam”solutions of the Schrodinger equation).


In the remaining cases where

|x − x∗| � Rθ

(note that there are O(θ3) such cases), we use the crude bound

|I(x)| � 1,

and obtain the final estimate( ∑x∈ R

100Z3

I(x)100)1/100

� θ3/100

as desired. Now consider the case θ � 1. In the region

|x − x∗| R

one can get very good bounds from nonstationary phase again, namely (14.5).There are only O(1) remaining values of x. For each of these values we ob-serve that the double derivative ∇2

zΦx is nondegenerate, indeed it is equalto 2R2 s−−s+

(t−s−)(t−s+) = 2θ times the identity matrix. Thus by the principle of

stationary phase (see [41]) we have

|I(x)| � θ3/2

in these cases. Upon summing we obtain( ∑x∈ R

100Z3

I(x)100)1/100

� θ3/2

as claimed.

Using Lemma 14.1, we can estimate the left-hand side of (14.4) as

�∫ ∫ ∫

t−<s−<t<s+<t+

1R|s+ − s−|−3/2

· min

(( |t − s+||t − s−|R2|s+ − s−|

)−3/2+3/100

, 1

)‖G(s−)‖L1

x‖G(s+)‖L1

xds−ds+dt.

(14.6)

Now we use the crucial time ordering s− < t < s+. An elementary computation(treating the cases s+ − s− < R2 and s+ − s− ≥ R2 separately) shows that

1R

∫s−<t<s+

|s+ − s−|−3/2 min

(( |t − s+||t − s−|R2|s+ − s−|

)−3/2+3/100

, 1

)dt

� min(R|s+ − s−|−3/2, R−1|s+ − s−|−1/2).


The kernel min(R|s|−3/2, R−1|s|−1/2) has an L1s norm of35 O(1). Thus by

Young’s inequality in time (and Cauchy-Schwarz in time), we can bound (14.6)by

‖G‖2L2

tL1x(I0×R3)

which is O(C0η1) by Proposition 12.2, as desired. This (finally!) concludes theproof of Proposition 4.9.

15. Preventing energy evacuation

We now prove Proposition 4.15. By the scaling (1.3) we may take Nmin =1.

15.1. The setup and contradiction argument. Since the N(t) can onlytake values in a discrete set (the integer powers of 2), there thus exists a timetmin ∈ I0 such that

N(tmin) = Nmin = 1.

At this time t = tmin, we see from (4.11) and (1.18) that we have a substantialamount of mass36 (and energy) at medium frequencies:

‖Pc(η0)≤·≤C(η0)u(tmin)‖L2(R3) ≥ c(η0).(15.1)

This should be contrasted with (4.5), which shows that there is not much massat the frequencies much higher than C(η0).

Our task is to prove (4.24). Suppose for contradiction that this estimatefailed; then there exists a time tevac ∈ I0 for which N(tevac) C(η5). If C(η5)is sufficiently large, we then see from Corollary 4.4 that energy has been almostentirely evacuated from low and medium frequencies at time tevac:

‖P<1/η5u(tevac)‖H1(R3) ≤ η5.(15.2)

Under the intuition that the L2 mass density does not rapidly adjust during theNLS evolution, (15.2) will be contradicted if the low frequency L2 mass (15.1)sticks around Nmin until tevac. We will validate this slow L2 mass motionintuition by proving a frequency localized L2 mass almost conservation lawwhich leads to the contradiction, provided that the ηj are chosen small enough.

35It is crucial to note here that the powers of R have cancelled out. This seems to bea consequence of dimensional analysis, although the presence of the frequency localization|ξ| � 1 makes this analysis heuristic rather than rigorous.

36Note that we are not using the assumption that u is Schwartz (and thus has finite L2

norm) to obtain these estimates; the bounds here are independent of the global L2 normof u, which may be very large even for fixed energy E because the very low frequenciescan simultaneously have small energy and large mass, and are also not preserved by thescale-invariance which we have exploited to normalize Nmin = 1.


Having surveyed the argument, we carry out the details. Fix tevac; by timereversal symmetry we may assume that tevac > tmin. (From (15.1) it is clearthat tevac cannot equal tmin).

From (15.1) we have

‖u>c(η0)(tmin)‖L2(R3) ≥ η1

(for instance). In particular, if

Phi := P≥η1004

, Plo := P<η1004

, uhi := Phiu, ulo := Plou,

then we have

‖uhi(tmin)‖L2(R3) ≥ η1.(15.3)

Suppose we could show that

‖uhi(tevac)‖L2(R3) ≥12η1.(15.4)

From (4.3) and (4.5) we would thus have

‖P≤C(η1)uhi(tevac)‖L2(R3) ≥14η1.

By (1.16) this implies that

‖P≤C(η1)u(tevac)‖H1(R3) � c(η1, η4).

But then Corollary 4.4 implies that N(tevac) � C(η1, η4), which contradicts(15.2) if η5 is chosen sufficiently small.

It remains to prove (15.4). We use the continuity method. Suppose wehave a time tmin ≤ t∗ ≤ tevac for which

inftmin≤t≤t∗

‖uhi(t)‖L2(R3) ≥12η1.(15.5)

We will show that (15.5) can be bootstrapped to

inftmin≤t≤t∗

‖uhi(t)‖L2(R3) ≥34η1.(15.6)

This implies that the set of times t∗ for which (15.5) holds is both open andclosed in [tmin, tevac], which will imply (15.4) as desired. Note that the intro-duction of η100

4 � Nmin allows for the L2 mass near Nmin at time tmin to movetoward low frequencies but the estimate (15.6) shows a portion of the massstays above η100

4 for t ∈ [tmin, tevac]. Note also that upon proving (15.6), wewill have established Lemma 4.14.

It remains to derive (15.6) from (15.5). The idea is to treat the L2 normof uhi(t); i.e.,

L(t) :=∫

R3

|uhi(t, x)|2 dx,


as an almost conserved quantity. From (15.3) we have

L(tmin) ≥ η21,

and so by the Fundamental Theorem of Calculus it will suffice to show that∫ t∗

tmin

|∂tL(t)| dt ≤ 1100

η21.

From (2.4) and (2.6) we have

∂tL(t) = 2∫

{Phi(|u|4u), uhi}m dx

= 2∫

{Phi(|u|4u) − |uhi|4uhi, uhi}m dx.

Thus it will suffice to show that∫ t∗

tmin

∣∣∣ ∫{Phi(|u|4u) − |uhi|4uhi, uhi}m dx

∣∣∣dt ≤ 1100

η21.(15.7)

The proof of (15.7) is accomplished using a quintilinear analysis of variousinteractions using three inputs: S1 Strichartz estimates on low frequencies, L4

xt

estimates via frequency localized interaction Morawetz and Bernstein estimateson medium and higher frequencies, and S1 Strichartz estimates on very highfrequencies.

15.2. Spacetime estimates for high, medium, and low frequencies. To prove(15.7) we need a number of spacetime bounds on u, which we now discuss inthis subsection. Observe by the discussion following (15.4) that the hypothesis(15.5) implies in particular that

N(t) ≤ C(η1, η4) for all tmin ≤ t ≤ t∗.

This in turn can be combined with Corollary 4.13 to obtain the useful Strichartzbounds

‖u‖S1([tmin,t∗]×R3) ≤ C(η1, η3, η4).(15.8)

Because of the dependence of the right-hand side on η4, this bound is onlyuseful for us when there is a power of η5 present. In all other circumstances,we resort instead to Proposition 4.9, which gives the L4

t,x bounds37∫ tevac

tmin

∫|P≥Nu(t, x)|4 dxdt � η1N

−3(15.9)

37It is important here to note that while the L2 control on uhi only extends from tmin upto t∗, the L4

t,x control on u≥N extends all the way up to tevac. This allows us to access theevacuation hypothesis (15.2) to provide useful new control, especially at low frequencies, inthe time interval [tmin, t∗]. This additional control will be crucial for us to obtain the desiredalmost conservation law on the mass of uhi, thus closing the bootstrap and allowing t∗ toextend all the way up to tevac, at which point we can declare a contradiction.


whenever N < c(η3). Roughly speaking, the bound (15.9) is better than (15.8)for medium frequencies, but (15.8) is superior for very high frequencies, andLemma 15.1 below will be superior for very low frequencies.

It turns out that we also need some bounds on the low frequencies suchas u≤η4 ; the estimate (15.9) is inappropriate for this purpose because N−3

diverges as N → 0. One can modify Proposition 12.1 to obtain some reasonablecontrol, but it turns out that the constants obtained by that estimate are notstrong enough to counteract the losses arising from (15.8). We need a strongerversion of Proposition 12.1 which takes advantage of the evacuation hypothesis(15.2), which asserts among other things that u≤η4 has extremely small energyat time tevac. Because of this hypothesis we expect u≤η4 to have extremelysmall energy at all other times in [tmin, tevac] (i.e. we expect bounds gaining anη5 instead of just an η3). Of course there is a little bit of energy leaking fromthe high frequencies to the low, but fortunately the L4

t,x bound on the highfrequencies will limit38 how far the high frequency energy can penetrate to thevery low modes. This intuition is made rigorous as follows:

Lemma 15.1. With the above notation and assumptions (in particular,assuming (15.2) and (15.9)) we have

‖P≤Nu‖S1([tmin,tevac]×R3) � η5 + η−3/24 N3/2(15.10)

for all N ≤ η4.

One should think of the Cη5 term on the right-hand side of (15.10) asthe energy coming from the low modes of u(tevac), while the η

−3/24 N3/2 term

comes from the nonlinear corrections generated by the high modes of u(t)for tmin ≤ t ≤ tevac. Note the very strong decay of N3/2 as N → 0; thismeans that the high modes cannot project their energy very far into the lowmodes. This estimate should be compared with Proposition 12.1. It beginsto deteriorate if N gets too close to η4; we avoid this problem by making thehigh-low frequency decomposition u = uhi + ulo at η100

4 instead of η4. Notethat this bound is superior to either (15.8) or (15.9) at low frequencies.

Proof. As usual, we rely on the continuity method, although now we willevolve39 backwards in time from tevac, rather than forwards from tmin. Let C0

38It may seem surprising that the L4t,x bound, which is supercritical, can lead to control on

critical quantities such as the energy. The point is that once one localizes in frequency, thedistinctions between subcritical, critical, and supercritical quantities become less relevant, asone can already see from Bernstein’s inequality (1.20). In this section the entire analysis islocalized around the frequency Nmin = 1, so that supercritical norms (such as L4

t,x or L∞t L2

x)can begin to play a useful role.

39The arguments in this section seem to rely in an essential way on evolving both forwardsand backwards in time simultaneously; compare with the “double Duhamel trick” in Section14.


be a large absolute constant (not depending on any of the ηj) to be chosenlater. Let Ω ⊆ [tmin, tevac) denote the set of all times tmin ≤ t < tevac such thatwe have the bounds

‖P≤Nu‖S1([t,tevac]×R3) ≤ C0η5 + η0η−3/24 N3/2(15.11)

for all N ≤ η4. To show (15.10), it will clearly suffice to show that tmin ∈ Ω (theadditional factor of η0 is useful for the continuity method but will be discardedfor the final estimate (15.10)). In particular, we observe from (15.11) that wehave

‖P≤Nu‖S1([t,tevac]×R3) � η0(15.12)

for all N ≤ η4.First we show that t ∈ Ω for t sufficiently close to tevac. We use (3.1),

Holder and Sobolev to estimate

‖P≤Nu‖S1([t,tevac]×R3) � ‖∇P≤Nu‖L∞t L2

x([t,tevac]×R3) + ‖∇u‖L2t L6

x([t,tevac]×R3)

� ‖∇P≤Nu(tevac)‖L2 + C|tevac − t|‖∇∂tu‖L∞t L2

x(I0×R3)

+ |tevac − t|1/2‖∇u‖L∞t L6

x(I0×R3).

Since u is Schwartz, the latter two norms are finite (though perhaps very large).By (15.2) we thus have the estimate

‖P≤Nu‖S1([t,tevac]×R3) � η5 + C(I0, u)|tevac − t| + C(I0, u)|tevac − t|1/2.

We now see that the bound (15.11) holds for t sufficiently close to tevac, if C0

is chosen large enough (but not depending on I0, u or any of the ηj .)Now suppose that t ∈ Ω, so that (15.11) holds for all N ≤ η4. We shall

bootstrap (15.11) to

‖P≤Nu‖S1([t,tevac]×R3) ≤12C0η5 +

12η0η

−3/24 N3/2(15.13)

for all N ≤ η4. If this claim is true, this would imply (since u is Schwartz)that Ω is both open and closed, and so we have tmin ∈ Ω as desired.

It remains to deduce (15.13) from (15.11). For the rest of the proof, allspacetime norms will be restricted to [t, tevac] × R3.

Fix N ≤ η4. By (1.1) and Lemma 3.2,

‖P≤Nu‖S1([t,tevac]×R3) � ‖P≤Nu(tevac)‖H1(R3) + CM∑

m=1

‖∇Fm‖L

q′mt L

r′mx ([t,tevac]×R3)

for some dual L2-admissible exponents (q′m, r′m), and some decompositionP≤N (|u|4u) =

∑Mm=1 Fm which we will give shortly.

From (15.2) we have

‖P≤Nu(tevac)‖H1(R3) � η5,

which is acceptable for (15.13) if C0 is large enough.


Now consider the nonlinear term P≤N (|u|4u). We split u into high and lowfrequencies u = u≤η4 +u>η4 , where of course u≤η4 := P≤η4u and u>η4 := P>η4u,and use (1.15) to decompose

P≤N (|u|4u) =5∑

j=0

Fj ,

where Fj := P≤NO(uj>η4

u5−j≤η4

). We now treat the various terms separately.

• Case 1. Estimation of F2, F3, F4, F5. We estimate these terms in L2t L

6/5x

norm. We use Bernstein’s inequality (1.19) to bound these terms by

CN3/2‖O(uj>η4

u5−j≤η4

)‖L2t L1

x([t,tevac]×R3),

which by Holder can be estimated by

CN3/2‖u>η4‖j−2L∞

t L6x([t,tevac]×R3)‖u≤η4‖5−j

L∞t L6

x([t,tevac]×R3)‖u>η4‖2L4

tL4x([t,tevac]×R3).

Applying (4.4) and Sobolev, as well as (15.9), we obtain a bound of

Cη121 N3/2η

−3/24 ,

which is acceptable for (15.13) if η1 is sufficiently small. It is at this step thatthe small constant η1 appearing in 4.19 is used to close the bootstrap.

• Case 2a. Estimation of F1 when N � η4. Now consider F1 term

‖∇P≤NO(u>η4u4≤η4

)‖L

q′1t L

r′1x

.(15.14)

Suppose first that N < cη4. Then this expression vanishes unless at least oneof the four u≤η4 factors has frequency > cη4. Thus we can essentially write(15.14) as40

‖∇P≤NO(u3≤η4

(P>cη4u≤η4)u>η4)‖L2t L

6/5x ([t,tevac]×R3),

where we have chosen (q′1, r′1) = (2, 6/5). By (15.9), the function P≥cη4u≤η4

obeys essentially the same L4t,x estimates as u>η4 , and so this term is acceptable

when we repeat the arguments used to deal with F2, F3, F4, F5.

• Case 2b. Estimation of F1 when N ∼ η4. Considering the case whenN ≥ cη4, we choose (q′1, r

′1) = (1, 2) and use (1.18) to bound (15.14) by

Cη4‖O(u>η4u4≤η4

)‖L1t L2

x([t,tevac]×R3)

40Strictly speaking, one can only write (15.14) as a sum of such terms, where some of theu≤η4 factors in u3

≤η4must be replaced by either P>cη4u≤η4 or P≤cη4u≤η4 . As these projections

are bounded on every space under consideration we ignore this technicality. Similarly forother such decompositions in this lemma.


which we estimate using Holder by

Cη4‖u≤η4‖4L4

t L∞x ([t,tevac]×R3)‖u>η4‖L∞

t L2x([t,tevac]×R3).

From (4.3) we obtain

‖u>η4‖L∞t L2

x([t,tevac]×R3) ≤ Cη−14 ,

while from (3.4) and (15.12) we obtain

‖u≤η4‖L4t L∞

x ([t,tevac]×R3) � ‖u≤η4‖S1([t,tevac]×R3) � η0.

Thus we can bound (15.14) by O(η40), which is acceptable for (15.13) since

N ∼ η4. This concludes the estimation of F1.

• Case 3. Estimation of F0. It remains to consider the F0 term; we set(q′0, r

′0) = (1, 2) and estimate


)‖L1t L2

x([t,tevac]×R3).

We split u≤η4 = u<η5 + uη5≤·≤η4 , and consider any term which involves thevery low frequencies u<η5 ; schematically, this is


u<η5)‖L1t L2

x([t,tevac]×R3).

For this case we discard the P≤N , and apply Lemma 3.6 to estimate this termby

� ‖u≤η4‖4S1([t,tevac]×R3)

‖u<η5‖S1([t,tevac]×R3).

Applying (15.11) we can bound this by

� (C0η5 + η0)4(C0η5 + η0η−3/24 η

3/25 ) � C0η

40η5

which is acceptable.We can thus discard all the terms involving u<η5 , and reduce to estimating

‖∇P≤NO(u5η5≤·≤η4

)‖L1t L2

x([t,tevac]×R3).(15.15)

By Bernstein (1.19), (1.17) we may estimate this by

� N3/2‖O(u5η5≤·≤η4

)‖L1t L

3/2x ([t,tevac]×R3) = N3/2‖uη5≤·≤η4‖5

L5t L

15/2x ([t,tevac]×R3)

.

But from (3.4), Bernstein (1.20), (1.18), and (15.11) we have

‖uη5≤·≤η4‖L5tL

15/2x

≤∑

η5≤N ′≤η4

‖PN ′u‖L5t L

15/2x ([t,tevac]×R3)

�∑

η5≤N ′≤η4

(N ′)−3/10‖∇PN ′u‖L5tL

30/11x ([t,tevac]×R3)

�∑

η5≤N ′≤η4

(N ′)−3/10‖PN ′u‖S1([t,tevac]×R3)

�∑

η5≤N ′≤η4

(N ′)−3/10(C0η5 + η0η−3/24 (N ′)3/2)

� η0η−3/104 ,


so that (15.15) is estimated by O(η50η

−3/24 N3/2) which is acceptable if η0 is

small enough. This proves (15.13), which closes the bootstrap.

15.3. Controlling the localized L2 mass increment. Now we have enoughestimates to prove (15.7). We first rewrite

Phi(|u|4u) − |uhi|4uhi =Phi(|u|4u − |uhi|4uhi − |ulo|4ulo)

+Phi(|ulo|4ulo) − Plo(|uhi|4uhi).

Now, it will suffice to consider the three quantities∫ t∗

tmin

|∫

uhiPhi(|u|4u − |uhi|4uhi − |ulo|4ulo) dx|dt,(15.16) ∫ t∗

tmin

|∫

uhiPhi(|ulo|4ulo) dx|dt,(15.17) ∫ t∗

tmin

|∫

uhiPlo(|uhi|4uhi) dx|dt,(15.18)

which we shall estimate below: (15.18).

• Case 1. Estimation of (15.16). We move the self-adjoint operator Phi

onto uhi, and then apply the pointwise estimate (cf. (1.15))

||u|4u − |uhi|4uhi − |ulo|4ulo| � |uhi|4|ulo| + |uhi||ulo|4

to bound (15.16) by

(15.16) �∫ t∗

tmin

∫|Phiuhi|(|uhi|4|ulo| + |uhi||ulo|4) dxdt.

For notational convenience, we will ignore the Phi projection and write Phiuhi

as uhi; strictly speaking this is not quite accurate but as Phiuhi obeys all thesame estimates as uhi, and we have already placed absolute values everywhere,this is a harmless modification. We can now write our bound for (15.16) as

(15.16) �∫ t∗

tmin

∫|uhi|5|ulo| + |uhi|2|ulo|4 dxdt.(15.19)

• Case 1a. Contribution of |uhi|2|ulo|4. Consider first the contributionof |uhi|2|ulo|4; we have to show that∫ t∗

tmin

∫|uhi|2|ulo|4dxdt � η2

1.(15.20)

By Holder we can bound this contribution by

(15.20) � ‖uhi‖2L∞

t L2x([tmin,t∗]×R3)‖ulo‖4

L4t L∞

x ([tmin,t∗]×R3).

From (15.10) we have

‖ulo‖L4t L∞

x ([tmin,t∗]×R3) � η3/24


while from (4.3), (15.10) we have

‖uhi‖L∞t L2

x([tmin,t∗]×R3) ≤ ‖P≥η4uhi‖L∞t L2

x([tmin,t∗]×R3)

+∑

η1004 ≤N≤η4

‖PNuhi‖L∞t L2

x([tmin,t∗]×R3)

� η−14 ‖u‖L∞

t H1x([tmin,t∗]×R3)

+∑

η1004 ≤N≤Cη4

N−1‖PN∇uhi‖L∞t L2

x([tmin,t∗]×R3)

� η−14 +

∑η1004 ≤N≤η4

N−1‖PNuhi‖S1([tmin,t∗]×R3)

� η−14 +

∑η1004 ≤N≤η4

N−1(η5 + η−3/24 N3/2)

� η−14 + η5η

−1004 + η−1

4

� η−14 .

We thus obtain a bound of O(η44), which is acceptable.

• Case 1b. Contribution of |uhi|5|ulo|. It remains to control the contri-bution of |uhi|5|ulo|; in other words, we need to show∫ t∗

tmin

∫|uhi|5|ulo|dxdt � η2

1.(15.21)

This estimate will also be useful in controlling (15.18).We will split ulo further into somewhat low, and very low, frequency pieces:

ulo = P>η1/25

ulo + P≤η1/25

u.

The contribution of the very low frequency piece to (15.21) can be boundedby Sobolev embedding (1.19) and (4.4) as∫ t∗

tmin

∫|uhi|5|u≤η

1/25

|dxdt

� ‖uhi‖5L5

t L5x([tmin,t∗]×R3)‖u≤η

1/25

‖L∞t L∞

x ([tmin,t∗]×R3)

� C(η4)‖∇u‖5L5

t L30/11x ([tmin,t∗]×R3)

η1/45 ‖∇u‖L∞

t L2x([tmin,t∗]×R3)

� C(η4)η1/45 ‖u‖5

S1([tmin,t∗]×R3),

which is acceptable by (15.8). Hence we only need to consider the somewhatlow frequencies P>η

1/25

ulo. By Holder, we obtain the bound

(15.22) ‖|uhi|5|P>η1/25

ulo|‖L1t,x([tmin,t∗]×R3)

≤ C‖uhi‖5L10

t L5x([tmin,t∗]×R3)‖P>η

1/25

ulo‖L2t L∞

x ([tmin,t∗]×R3).


From Bernstein (1.20) and (15.10) we have

‖P>η1/25

ulo‖L2t L∞

x ([tmin,t∗]×R3) ≤∑

η1/25 <N≤η100

4

‖PNu‖L2t L∞

x ([tmin,t∗]×R3)

≤∑

η1/25 <N≤η100

4

N−1/2‖∇PNu‖L2t L6

x([tmin,t∗]×R3)

≤∑

η1/25 <N≤η100

4

N−1/2‖PNu‖S1([tmin,t∗]×R3)

≤∑

η1/25 <N≤η100

4

N−1/2(η5 + η−3/24 N3/2)

≤ Cη−3/24 η100

4 .

To estimate ‖uhi‖L10t L5

x([tmin,t∗]×R3), we split uhi into the higher frequencies u>η4

and the medium frequencies uη1004 ≤·≤η4

. For the higher frequencies we use41

(15.9), (4.4), and Holder to obtain

‖u>η4‖L10t L5

x([tmin,t∗]×R3) � ‖u>η4‖2/5L4

t L4x([tmin,t∗]×R3)‖u>η4‖

3/5L∞

t L6x([tmin,t∗]×R3)

� η−3/104 ,

while for the medium frequencies we instead use Bernstein (1.20), (1.18), (3.4)and Lemma 15.1 to estimate

‖uη1004 ≤·≤η4

‖L10t L5

x([tmin,t∗]×R3) �∑

η1004 ≤N≤η4

‖uN‖L10t L5

x([tmin,t∗]×R3)

�∑

η1004 ≤N≤η4

N−3/10‖∇uN‖L10t L

30/13x ([tmin,t∗]×R3)

�∑

η1004 ≤N≤η4

N−3/10‖uN‖S1([tmin,t∗]×R3)

�∑

η1004 ≤N≤η4

N−3/10(η5 + η−3/24 N3/2)

� η−3/104 .

Inserting these bounds into (15.22) we obtain a bound of η−34 η100

4 for (15.21),which is acceptable.

• Case 2. Estimation of (15.17). Because of the presence of the Phi

projection, one of the ulo terms must have frequency ≥ cη1004 . We then move

Phi over to the uhi, bounding (15.17) as a sum of terms which are essentially

41Note that this application of (15.9) does not require the small constant η1.


of the form42 ∫ t∗

tmin

∫|Phiuhi||P≥cη100

4ulo||ulo|4 dxdt.

Now observe that Phiuhi and P≥cη1004

ulo = Plou≥cη1004

satisfy essentially thesame estimates as uhi, so that this expression can be shown to be acceptableby a minor modification of (15.20).

• Case 3. Estimation of (15.18). We move projections around, using theidentity Plouhi = Phiulo, to write (15.18) as∫ t∗

tmin

|∫

Phiulo|uhi|4uhidx|dt.(15.23)

Thus, we are concerned here with a term involving five uhi factors and onePhiulo factor. But this is basically (15.21), which has already been shown tobe acceptable. (We have Phiulo instead of ulo but the reader may verify thatthe Phi is harmless since it does not destroy any of the estimates of ulo).

This proves (15.7), and the proof of Proposition 4.15 is complete. This(finally!) concludes the proof of Theorem 1.1.

16. Remarks

We make here some miscellaneous remarks concerning certain variants ofTheorem 1.1.

The global well-posedness result in Theorem 1.1 was asserted with regardto finite energy solutions u in the class C0

t H1x∩L10

t,x, in that the solution existedand was unique in this class, and depended continuously on the initial data(cf. Lemma 3.10). However, the uniqueness result can be strengthened, in thesense that the solution constructed by Theorem 1.1 is in fact the only suchsolution in the class C0

t H1x (without the assumption of finite L10

t,x norm). Thistype of “unconditional well-posedness” result was first obtained in [26], [27](see also [15], [14]); the result in [26], [27] was phrased for the sub-criticalSchrodinger equation but can be extended to the critical setting thanks to theendpoint Strichartz estimates in [32] (or Lemma 3.2). For the convenience ofthe reader we sketch here the ideas of this argument, which are essentiallyin [27], [15], [14], we are indebted to Thierry Cazenave for pointing out therelevance of the endpoint Strichartz estimate to the H1-critical uniquenessproblem.

Let u0 be finite energy initial data, and let u ∈ C0t H1

x∩L10t,x be the (global)

solution to (1.1) constructed in Theorem 1.1 with these initial data; thus u(0)= u0. Suppose for contradiction that we have another (local or global) solution

42Actually, some of the ulo factors in |ulo|4 may have to be replaced by either P≥cη1004

ulo

or P<cη1004

ulo, but this will make no difference to the estimates.


v ∈ C0t H1

x to (1.1) with initial u0, in the sense that v verifies the (Duhamel)integral formulation of (1.1),

v(t) = eitΔu0 − i

∫ t

0ei(t−s)Δ(|v|4v(s)) ds.

Note that v ∈ C0t H1

x ⊆ C0t L6

x by Sobolev embedding, so that, in particular,the nonlinearity |v|4v is locally integrable, and the right-hand side of the aboveformula makes sense distributionally at least. We now claim that u ≡ v onthe entire time interval for which v is defined. Actually, we shall just showthat u ≡ v for all times t in a sufficiently small neighborhood I of 0; one canthen extend this to the whole time interval by a continuity argument and timetranslation invariance.

To prove the claim, we write v = u + w and observe that w obeys adifference equation, which we write in integral form as

w(t) = −i

∫ t

0ei(t−s)Δ(|u + w|4(u + w)(s) − |u|4u(s)) ds.

Let ε > 0 be a small number to be chosen shortly. Note that w ∈ C0t H1

x ⊆ C0t L6

x

and w(0) = 0, so that in particular we can ensure that ‖w‖L∞t L6

x(I×R3) ≤ ε bychoosing I sufficiently small. Also, from the Strichartz analysis u has finiteS1 norm, and in particular it has finite L8

t L12x norm. Thus we can also ensure

that ‖u‖L8tL12

x (I×R3) ≤ ε by choosing I sufficiently small. Now we use (1.15) towrite the equation for w as

w(t) =∫ t

0ei(t−s)Δ(O(|w(s)|5) + O(|u(s)|4|w(s)|)) ds.

We apply Lemma 3.2 with k = 0 to conclude in particular that

‖w‖L2t L6

x(I×R3) ≤ C‖|w|5‖L2t L

6/5x (I×R3) + C‖|u|4|w|‖L1

tL2x(I×R3).

From our choice of I and Holder’s inequality we see in particular that

‖w‖L2t L6

x(I×R3) ≤ Cε4‖w‖L2tL6

x(I×R3).

Note that the L2t L

6x(I × R3) norm of w is finite since w ∈ C0

t L6x. If we choose

ε sufficiently small, we then conclude that w vanishes identically on I × R3.One can then extend this vanishing to the entire time interval for which v isdefined by a standard continuity argument which we omit.

We now briefly discuss possible extensions to Theorem 1.1. One obviousextension to study would be the natural analogue of Theorem 1.1 in higherdimensions n > 3, with the equation (1.1) replaced by its higher-dimensionalenergy-critical counterpart

iut + Δu = |u|4

n−2 u.


The four-dimensional case n = 4 seems particularly tractable since the nonlin-earity is cubic.43 In higher dimensions n ≥ 5 one no longer expects a regularityresult since the nonlinearity is not smooth when u vanishes. However one mightstill hope for a global well-posedness result in the energy space (especially sincethis is already known to be true for small energies; see [9]). In the radial case,such a result was obtained in four dimensions in [4], [5] and more recently ingeneral dimension in [45], and so it is reasonable to conjecture that one infact has global well-posedness in the energy space for all dimensions n ≥ 3and all finite energy data, in analogy with Theorem 1.1. However, extendingour arguments here to the higher dimensional setting is far from automatic,even in the four-dimensional case; all the Strichartz numerology changes, ofcourse, but also the interaction Morawetz inequality behave in a somewhatdifferent manner in higher dimensions (since the quantity Δ 1

|x| is no longera Dirac mass, but instead a fractional integral potential). However, it seemsthat other parts of the argument, such as the induction on energy machinery,the localization of minimal-energy blowup solutions, and the energy evacuationarguments based on frequency-localized approximate mass conservation laws,do have a good chance of extending to this setting. We will not pursue thesematters in detail here.

Another natural extension would be to add a lower order nonlinearity to(1.1), for instance combining the pure power quintic nonlinearity |u|4u witha pure power cubic nonlinearity |u|2u. Heuristically, we do not expect suchlower order terms to affect the global well-posedness and regularity of theequation (especially if those terms have the same defocusing sign as the toporder term), although they may cause some difficulty in obtaining a scatteringresult (especially if one adds a nonlinearity of the form |u|p−1u for very low p,such as p ≤ 1 + 4

n = 73 or p ≤ 1 + 2

n = 53). However, these lower order terms

do create some nontrivial difficulties in our argument, which relies heavily onscale-invariance. One may need to add some lower order terms (such as the L2

mass) to the energy E, or to the definition of the quantity M(E), in order tosalvage the induction on energy argument in this setting. Again, we will notpursue these matters here.44

As remarked in Remark 5.3, our final bound M(E) for the global L10t,x

norm of u in terms of the energy E is extremely bad; this is due to our ex-tremely heavy reliance on the induction on energy hypothesis (Lemma 4.1) in

43Note added in proof: The four-dimensional case has been handled by a very recentpreprint of Ryckman and Visan [37], using a modification of the methods here. The case ofdimensions five and higher has also been very recently settled (Visan, personal communica-tion).

44Note added in proof: the lower order terms have been successfully treated by XiaoyiZhang (personal communication), relying on this result and perturbation theory.


our argument.45 We do not expect our bounds to be anywhere close to bestpossible. Indeed, any simplification of this argument would almost surely leadto less use of the induction hypothesis, and consequently to a better boundon M(E). For recent progress in this direction in the radial case (in whichno induction hypothesis is used at all, leading to a bound on M(E) which ismerely exponential in E), see [45].

The global existence and scattering result obtained here has analogs for thecritical nonlinear Klein-Gordon equation − 1

2c2 utt + Δu = −|u|4u + m2c2

2 u (seeintroduction for references). As we remarked earlier, there are some importantdifferences between the methods employed for the Klein-Gordon equation andthose we use here. In particular, it is not at all clear how our arguments mighthelp show that the space-time bounds for the nonlinear Klein-Gordon equationare uniform in the nonrelativistic limit c → ∞, even though one heuristicallyexpects the nonlinear Klein-Gordon equation to converge in some sense to thenonlinear Schrodinger equation in this regimen with suitable normalizationsand assumptions on the data. One major difficulty in extending our argu-ments to the relativistic case is that we have no analogue of the interactionMorawetz inequality (1.8) (or any localized variants) for the Klein-Gordonequation. For small energy data, uniform bounds on the solution are availablein the nonrelativistic limit (see remarks in [31]), but for general solutions suchbounds do not seem available. (See also [29] and references therein for furtherresults on the subcritical problem.)

University of Toronto, Toronto, Ontario, Canada

E-mail address: [email protected]

University of Minnesota, Minneapolis, MN


Massachusetts Institute of Technology, Cambridge, MA


Kobe University, Kobe, Japan


University of California, Los Angeles, and The Australian National Univer-

sity, Canberra, Australia


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(Received February 10, 2004)

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Global well-posedness and scattering for the energy-critical